User:לערי ריינהארט/sandbox

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User:לערי ריינהארט

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re: talk:Tree of primitive Pythagorean triples[edit]

from talk:Tree of primitive Pythagorean triples&direction=prev&oldid=906435953 Gangleri 217.86.249.144 (talk) 12:15, 16 July 2019 (UTC)

Dual representation for the tree[edit]

1 0 1
1 0
3 4 5
2 1
5 12 13
3 2
7 24 25
4 3
9 40 41
5 4
105 88 137
11 4
91 60 109
10 3
55 48 73
8 3
105 208 233
13 8
297 304 425
19 8
187 84 205
14 3
45 28 53
7 2
95 168 193
12 7
207 224 305
16 7
117 44 125
11 2
21 20 29
5 2
39 80 89
8 5
57 176 185
11 8
377 336 505
21 8
299 180 349
18 5
119 120 169
12 5
217 456 505
19 12
697 696 985
29 12
459 220 509
22 5
77 36 85
9 2
175 288 337
16 9
319 360 481
20 9
165 52 173
13 2
15 8 17
4 1
33 56 65
7 4
51 140 149
10 7
275 252 373
18 7
209 120 241
15 4
65 72 97
9 4
115 252 277
14 9
403 396 565
22 9
273 136 305
17 4
35 12 37
6 1
85 132 157
11 6
133 156 205
13 6
63 16 65
8 1


Hi! The table above combines the representation of the primitive Pythagorean triplets with the values of the primitive pair of integers generating them.
The table includes as additional root the trivial triplet (1, 0, 1) which is generated by the (m, n) pair (1, 0). Please note that all (m, n) pairs are written with m < n which is the opposite of the requirement described in the article.
The alternative calculations of the childs of any Pythagorean triplet are as follow:

up: (mnew, nnew) = (2m-n, m)
Pell pair: (mnew, nnew) = (2m+n, m)
for (m, n) other then (1, 0) down: (mnew, nnew) = (m+2n, n)
note: from (1, 0) both the up path and the "Pell" path generate the same main root value (2, 1)

The calculations of the triplet childs might be simpler as using matrix calculations. Please note that both the up way and the "Pell" way are preserving m, the greater value of the (m, n) pair, while the down way preserves n, the smaller value of the (m, n) pair.
Additional remark on the Pell numbers: The sequence consists of values of the hypotenuse of a Pythagorean triplet followed by its circumference (a+b+c). It starts with 1 and 2. This pair generates the triplet (3, 4, 5) so both 5 and 12 = 3 + 4 + 5 can be added to the sequence. By using (2, 5) one generates the triplet (21, 20, 29) so both 29 and 70 = 21 + 20 +29 can be added to the sequence. Nevertheless (5, 12) is the next pair to be used on the "Pell" path.
Best regards Gangleri

The triplet (0, 1, 1) generated by the (1, 0) pair does not have a parent Pythagorean triplet.
In order to calculate the parent of a distinct arbitrary primitive Pythagorean triplet (a, b, c) first one must identify the generating (m, n) pair which is generating the examined Pythagorean triplet. The pair is identified via
m=square root(c²-a²)/2 and because m is not zero for triples other then (0, 1, 1) n=b/2m
algorithm:
note: the case for n=0 is discussed above
if m-2n=0 then the analysed Pythagorean triplet is the main root Pythagorean triplet (3, 4, 5)
if m-2n<0 then (mnew, nnew) = (n, 2n-m) and the way was "up"
else if n>m-2n then (mnew, nnew) = (n, m-2n) and the way was the "Pell" way
else (mnew, nnew) = (m-2n, n) and the way was "down".
Regards Gangleri — Preceding unsigned comment added by 2A02:810D:41C0:134:5D5D:FEFD:1D74:D8AF (talk) 12:22, 6 July 2019 (UTC)

Switched to (m, n) with m > n representation. Regards Gangleri 13:46, 7 July 2019 (UTC)

Hopping on the Pell path[edit]

Any two consecutive numbers of the Pell series Pk = n and Pk + 1 = m generate a Pythagorean triplet (a, b, c). The (m, n) pair is used to generate (m² - n², 2mn, m² + n²) with the additional property that | a - b | = 1. Take p = 2m² + 2mn the circumference of the triangle and q = m² + n² the hypotenuse of the triangle.
With (p, q) a new Pythagorean triplet can be generated having an interesting circumference and an interesting hypotenuse.
Now consider the iteration pnew = 2p² + 2pq the circumference and qnew = p² + q² the hypotenuse. These are two different consecutive numbers of the Pell series so they generat a Pythagorean triplet with | a - b | = 1.
Examples:

(P2, P1) i.e. (2, 1) iterates to
(P4, P3) i.e. (12, 5) iterates to
(P8, P7) i.e. (408 169) iterates to
(P16, P15) i.e. (470832 195025) iterates to
(P32, P31) i.e. (627013566048 259717522849)

Best regards Gangleri 18:42, 7 July 2019 (UTC) — Preceding unsigned comment added by 2A02:810D:41C0:134:F81F:7483:C15A:1780 (talk)

What are neither coincidences nor obvious for any two consecutive numbers of the Pell series (Pn, Pn - 1) are properties for their generated Pythagorean triangle (a, b, c)
i) c is a Pell number
ii) a + b + c is a Pell number too
iii) a + b + c is the successor of c in the Pell series
Best regards Gangleri 20:33, 7 July 2019 (UTC)

Note that the predecessor of the Pell number c can be calculated directly as a + b - c. So (c, a + b - c) can be used as a pair of two consecutive Pell numbers witch differs from the original (m, n) if that was not (1, 0).
Because of this relation an additional hopping path can be used in order to find two consecutive Pell numbers. From Pk = n and Pk + 1 = m we use the hypotenuse c of the generated Pythagorean triangle and its predecessor a + b - c using the formulas

mnew = m² + n²
nnew = 2mn - 2n²

Examples:

(P2, P1) i.e. (2, 1) iterates to
(P3, P2) i.e. (5, 2) iterates to
(P5, P4) i.e. (29 12) iterates to
(P9, P8) i.e. (985 408) iterates to
(P17, P16) i.e. (1136689 470832)

Conclusion: Together with the original Pell iteration Pn = Pn - 1 + 2Pn - 2 (for n >=2 ) there are tree paths from one pair of consecutive Pell numbers other then (1, 0) to another pair of consecutive Pell numbers.
Best regards Gangleri 09:37, 8 July 2019 (UTC)

Trees and their objects[edit]

examples
Berggrens's Price's
0,113313
1643 924 1885
42 11
53 31
4452 682
1,22212
1643 924 1885
42 11
53 31
4452 682
0,131331
1235 1932 2293
42 23
65 19
5460 874
to be calculated
1235 1932 2293
42 23
65 19
5460 874
1,113213
to be calculated
1695 6272 6497
64 49
113 15
14464 1470
0,11123111
1,33233
1695 6272 6497
64 49
113 15
14464 1470
0,122222
23661 23660 33461
169 70 #Pell(7,6)
239 99
80782 13860
to be calculated
23661 23660 33461
169 70 #Pell(7,6)
239 99
80782 13860
1,11132111121

Each primitive Pythagorean triplet can be identified by one of the following key values:

a) generating method for an arbitrary primitive Pythagorean triplet and the path to that triplet; because only two exhausting methods are known this could be either one of
a1) Berggrens's method and the path to the triplet in the Berggrens's tree as "0,11213" used below
a2) Price's method and the path to the triplet in the Price's tree as "1,2232" used below
or both
b) triplet values as (a, b, c) where a and c are odd integer numbers and b is an even integer number which might be zero; gcd(a, b, c) = 1 and a, b and c fullfill the Pythagorean equation a² + b² = c²; triplets not obeing the Pythagorean equation are not discussed in the following
c) the generating pair (m, n) where m and n are integer numbers m > n >= 0 and gcd(m, n) = 1 generating the primitive Pythagorean triplet (a, b, c) with the set of formulas 1 ; any artitrary pair (m, n) with the properties from above identifies a unique primitive Pythagorean triplet
d) the generating pair (u, v) where u and v are integer numbers u > v and both values are odd and gcd(u, v) = 1 and they generate the same primitive Pythagorean triplet (a, b, c) with the set of formulas 2 ; any artitrary pair (m, n) with the properties from above identifies a unique primitive Pythagorean triplet
e) the pair (p, l) formed by the numerical values of the perimeter and "lead" for a triangle formed by a primitive Pythagorean triplet (a, b, c) ; the lead of the sum of the length of the two legs of the associated Pythagorean triangle to the length of the hypotenuse of the Pythagorean triangle;
please note the relations p = a + b + c and l = a + b - c as well as c = (p - l) / 2 ; both p and l are even numbers; there are pairs of (p, l) which do neiter relate to primitive Pythagorean triplet nor to general Pythagorean triplet; this means that (p, l) can not be selected randomly

The necessary calculations / formulas to determine the four other associated key values when one key is known will be described in the detail subparagraphs.
Notes: The perimeters of two triangles associated to two different Pythagorean triplets might have the same values as can be seen searching for the hashtags "#p1a" to "#p3a" and "#p1b" to "#p5b" in the following two tables.

Berggrens's tree[edit]

Berggrens's tree
all generations (0 to 6)
0
0
1 0 1
1 0 #Plato(0) #Pell(1,0)
n/a
2 0
0,1
1
3 4 5
2 1 #Plato(1) #Pythagoras(1) #Pell(2,1)
3 1
4 4
12 2
0,11
5 12 13
3 2 #Plato(2)
5 1
12 9
30 4
0,111
7 24 25
4 3 #Plato(3)
7 1
24 16
56 6
0,1111
9 40 41
5 4 #Plato(4)
9 1
40 25
90 8
0,11111
1,13
11 60 61
6 5 #Plato(5)
11 1
60 36
132 10
0,111111
1,31
13 84 85
7 6 #Plato(6)
13 1
84 49
182 12
0,111112
253 204 325
17 6
23 11
204 289
782 132
0,111113
231 160 281
16 5
21 11
160 256
672 110
0,11112
1,1212
171 140 221
14 5
19 9
140 196
532 90
0,111121
333 644 725
23 14
37 9
644 529
1702 252
0,111122
893 924 1285
33 14
47 19
924 1089
3102 532
0,111123
551 240 601
24 5
29 19
240 576
1392 190
0,11113
1,11211
153 104 185
13 4
17 9
104 169
442 72
0,111131
315 572 653
22 13
35 9
572 484
1540 234
0,111132
731 780 1069
30 13
43 17
780 900
2580 442
0,111133
425 168 457
21 4
25 17
168 441
1050 136
0,1112
1,3211
105 88 137
11 4
15 7
88 121
330 56
0,11121
1,3212
203 396 445
18 11
29 7
396 324
1044 154
0,111211
301 900 949
25 18
43 7
900 625
2150 252
0,111212
1885 1692 2533
47 18
65 29
1692 2209
6110 1044
0,111213
1,11222
1479 880 1721
40 11
51 29
880 1600
4080 638
0,11122
555 572 797
26 11
37 15
572 676
1924 330
0,111221
1,31221
1005 2132 2357
41 26
67 15
2132 1681
5494 780
0,111222
3293 3276 4645
63 26
89 37
3276 3969
11214 1924
0,111223
2183 1056 2425
48 11
59 37
1056 2304
5664 814
0,11123
345 152 377
19 4
23 15
152 361
874 120
0,111231
795 1292 1517
34 19
53 15
1292 1156
3604 570
0,111232
1403 1596 2125
42 19
61 23
1596 1764
5124 874
0,111233
713 216 745
27 4
31 23
216 729
1674 184
0,1113
91 60 109
10 3
13 7
60 100
260 42
0,11131
1,1221
189 340 389
17 10
27 7
340 289
918 140
0,111311
287 816 865
24 17
41 7
816 576
1968 238
0,111312
1,22122
1647 1496 2225
44 17
61 27
1496 1936
5368 918
0,111313
1,22321
1269 740 1469
37 10
47 27
740 1369
3478 540
0,11132
429 460 629
23 10
33 13
460 529
1518 260
0,111321
767 1656 1825
36 23
59 13
1656 1296
4248 598
0,111322
2607 2576 3665
56 23
79 33
2576 3136
8848 1518
0,111323
1749 860 1949
43 10
53 33
860 1849
4558 660
0,11133
247 96 265
16 3
19 13
96 256
608 78
0,111331
585 928 1097
29 16
45 13
928 841
2610 416
0,111332
969 1120 1481
35 16
51 19
1120 1225
3570 608
0,111333
475 132 493
22 3
25 19
132 484
1100 114
0,112
55 48 73
8 3
11 5
48 64
176 30
0,1121
1,12111
105 208 233
13 8
21 5
208 169
546 80
0,11211
1,12113
155 468 493
18 13
31 5
468 324
1116 130
0,112111
1,12131
205 828 853
23 18
41 5
828 529
1886 180
0,112112
2077 1764 2725
49 18
67 31
1764 2401
6566 1116
0,112113
1767 1144 2105
44 13
57 31
1144 1936
5016 806
0,11212
987 884 1325
34 13
47 21
884 1156
3196 546
0,112121
1869 3740 4181
55 34
89 21
3740 3025
9790 1428
0,112122
5405 5508 7717
81 34
115 47
5508 6561
18630 3196
0,112123
3431 1560 3769
60 13
73 47
1560 3600
8760 1222
0,11213
777 464 905
29 8
37 21
464 841
2146 336
0,112131
1659 2900 3341
50 29
79 21
2900 2500
7900 1218
0,112132
3515 3828 5197
66 29
95 37
3828 4356
12540 2146
0,112133
1961 720 2089
45 8
53 37
720 2025
4770 592
0,1122
297 304 425
19 8
27 11
304 361
1026 176
0,11221
539 1140 1261
30 19
49 11
1140 900
2940 418
0,112211
781 2460 2581
41 30
71 11
2460 1681
5822 660
0,112212
5341 4740 7141
79 30
109 49
4740 6241
17222 2940
0,112213
4263 2584 4985
68 19
87 49
2584 4624
11832 1862
0,11222
1755 1748 2477
46 19
65 27
1748 2116
5980 1026
0,112221
3213 6716 7445
73 46
119 27
6716 5329
17374 2484
0,112222
10205 10212 14437
111 46
157 65
10212 12321
34854 5980
0,112223
6695 3192 7417
84 19
103 65
3192 7056
17304 2470
0,11223
1161 560 1289
35 8
43 27
560 1225
3010 432
0,112231
2619 4340 5069
62 35
97 27
4340 3844
12028 1890
0,112232
4859 5460 7309
78 35
113 43
5460 6084
17628 3010
0,112233
2537 816 2665
51 8
59 43
816 2601
6018 688
0,1123
1,2112
187 84 205
14 3
17 11
84 196
476 66
0,11231
429 700 821
25 14
39 11
700 625
1950 308
0,112311
671 1800 1921
36 25
61 11
1800 1296
4392 550
0,112312
3471 3200 4721
64 25
89 39
3200 4096
11392 1950
0,112313
2613 1484 3005
53 14
67 39
1484 2809
7102 1092
0,11232
765 868 1157
31 14
45 17
868 961
2790 476
0,112321
1,12223
1343 2976 3265
48 31
79 17
2976 2304
7584 1054
0,112322
4815 4712 6737
76 31
107 45
4712 5776
16264 2790
0,112323
3285 1652 3677
59 14
73 45
1652 3481
8614 1260
0,11233
391 120 409
20 3
23 17
120 400
920 102
0,112331
969 1480 1769
37 20
57 17
1480 1369
4218 680
0,112332
1449 1720 2249
43 20
63 23
1720 1849
5418 920
0,112333
667 156 685
26 3
29 23
156 676
1508 138
0,113
45 28 53
7 2
9 5
28 49
126 20
0,1131
95 168 193
12 7
19 5
168 144
456 70
0,11311
1,2211
145 408 433
17 12
29 5
408 289
986 120
0,113111
1,2213
195 748 773
22 17
39 5
748 484
1716 170 #p1a
0,113112
1827 1564 2405
46 17
63 29
1564 2116
5796 986
0,113113
1537 984 1825
41 12
53 29
984 1681
4346 696
0,11312
817 744 1105
31 12
43 19
744 961
2666 456
0,113121
1539 3100 3461
50 31
81 19
3100 2500
8100 1178
0,113122
4515 4588 6437
74 31
105 43
4588 5476
15540 2666
0,113123
2881 1320 3169
55 12
67 43
1320 3025
7370 1032
0,11313
627 364 725
26 7
33 19
364 676
1716 266 #p1a
0,113131
1,23221
1349 2340 2701
45 26
71 19
2340 2025
6390 988
0,113132
2805 3068 4157
59 26
85 33
3068 3481
10030 1716
0,113133
1,21232
1551 560 1649
40 7
47 33
560 1600
3760 462
0,1132
1,322
207 224 305
16 7
23 9
224 256
736 126
0,11321
369 800 881
25 16
41 9
800 625
2050 288
0,113211
531 1700 1781
34 25
59 9
1700 1156
4012 450
0,113212
3731 3300 4981
66 25
91 41
3300 4356
12012 2050
0,113213
2993 1824 3505
57 16
73 41
1824 3249
8322 1312
0,11322
1265 1248 1777
39 16
55 23
1248 1521
4290 736
0,113221
2323 4836 5365
62 39
101 23
4836 3844
12524 1794
0,113222
7315 7332 10357
94 39
133 55
7332 8836
25004 4290
0,113223
4785 2272 5297
71 16
87 55
2272 5041
12354 1760
0,11323
851 420 949
30 7
37 23
420 900
2220 322
0,113231
1909 3180 3709
53 30
83 23
3180 2809
8798 1380
0,113232
3589 4020 5389
67 30
97 37
4020 4489
12998 2220
0,113233
1887 616 1985
44 7
51 37
616 1936
4488 518
0,1133
1,1121
117 44 125
11 2
13 9
44 121
286 36
0,11331
279 440 521
20 11
31 9
440 400
1240 198
0,113311
441 1160 1241
29 20
49 9
1160 841
2842 360
0,113312
2201 2040 3001
51 20
71 31
2040 2601
7242 1240
0,113313
1,22212
1643 924 1885
42 11
53 31
924 1764
4452 682
0,11332
455 528 697
24 11
35 13
528 576
1680 286
0,113321
793 1776 1945
37 24
61 13
1776 1369
4514 624
0,113322
2905 2832 4057
59 24
83 35
2832 3481
9794 1680
0,113323
1995 1012 2237
46 11
57 35
1012 2116
5244 770
0,11333
221 60 229
15 2
17 13
60 225
510 52
0,113331
559 840 1009
28 15
43 13
840 784
2408 390
0,113332
799 960 1249
32 15
47 17
960 1024
3008 510
0,113333
357 76 365
19 2
21 17
76 361
798 68
0,12
1,21
21 20 29
5 2 #Pell(3,2)
7 3
20 25
70 12
0,121
39 80 89
8 5
13 3
80 64
208 30
0,1211
57 176 185
11 8
19 3
176 121
418 48
0,12111
75 308 317
14 11
25 3
308 196
700 66
0,121111
93 476 485
17 14
31 3
476 289
1054 84
0,121112
1,32321
1325 1092 1717
39 14
53 25
1092 1521
4134 700
0,121113
1,32122
1175 792 1417
36 11
47 25
792 1296
3384 550
0,12112
779 660 1021
30 11
41 19
660 900
2460 418
0,121121
1501 2940 3301
49 30
79 19
2940 2401
7742 1140
0,121122
4141 4260 5941
71 30
101 41
4260 5041
14342 2460
0,121123
2583 1144 2825
52 11
63 41
1144 2704
6552 902
0,12113
665 432 793
27 8
35 19
432 729
1890 304
0,121131
1387 2484 2845
46 27
73 19
2484 2116
6716 1026
0,121132
3115 3348 4573
62 27
89 35
3348 3844
11036 1890
0,121133
1785 688 1913
43 8
51 35
688 1849
4386 560
0,1212
377 336 505
21 8
29 13
336 441
1218 208
0,12121
715 1428 1597
34 21
55 13
1428 1156
3740 546
0,121211
1053 3196 3365
47 34
81 13
3196 2209
7614 884
0,121212
6765 6052 9077
89 34
123 55
6052 7921
21894 3740
0,121213
5335 3192 6217
76 21
97 55
3192 5776
14744 2310
0,12122
2059 2100 2941
50 21
71 29
2100 2500
7100 1218
0,121221
3741 7900 8741
79 50
129 29
7900 6241
20382 2900
0,121222
12141 12100 17141
121 50
171 71
12100 14641
41382 7100
0,121223
8023 3864 8905
92 21
113 71
3864 8464
20792 2982
0,12123
1305 592 1433
37 8
45 29
592 1369
3330 464
0,121231
2987 4884 5725
66 37
103 29
4884 4356
13596 2146
0,121232
5355 6068 8093
82 37
119 45
6068 6724
19516 3330
0,121233
2745 848 2873
53 8
61 45
848 2809
6466 720
0,1213
299 180 349
18 5
23 13
180 324
828 130
0,12131
637 1116 1285
31 18
49 13
1116 961
3038 468
0,121311
975 2728 2897
44 31
65 13
2728 1936
6600 806
0,121312
5439 4960 7361
80 31
111 49
4960 6400
17760 3038
0,121313
4165 2412 4813
67 18
85 49
2412 4489
11390 1764
0,12132
1,32221
1357 1476 2005
41 18
59 23
1476 1681
4838 828
0,121321
2415 5248 5777
64 41
105 23
5248 4096
13440 1886
0,121322
8319 8200 11681
100 41
141 59
8200 10000
28200 4838
0,121323
5605 2772 6253
77 18
u 105
2772 5929
14630 2124
0,12133
759 280 809
28 5
33 23
280 784
1848 230
0,121331
1817 2856 3385
51 28
79 23
2856 2601
8058 1288
0,121332
2937 3416 4505
61 28
89 33
3416 3721
10858 1848
0,121333
0,12312
1419 380 1469
38 5
43 33
380 1444
3268 330
0,122
1,122
119 120 169
12 5 #Pell(4,3)
17 7
120 144
408 70
0,1221
1,21211
217 456 505
19 12
31 7
456 361
1178 168
0,12211
315 988 1037
26 19
45 7
988 676
2340 266
0,122111
413 1716 1765
33 26
59 7
1716 1089
3894 364
0,122112
4365 3692 5717
71 26
97 45
3692 5041
13774 2340
0,122113
3735 2432 4457
64 19
83 45
2432 4096
10624 1710
0,12212
2139 1900 2861
50 19
69 31
1900 2500
6900 1178 #p2a
0,122121
4061 8100 9061
81 50
131 31
8100 6561
21222 3100
0,122122
11661 11900 16661
119 50
169 69
11900 14161
40222 6900
0,122123
7383 3344 8105
88 19
107 69
3344 7744
18832 2622
0,12213
1705 1032 1993
43 12
55 31
1032 1849
4730 744
0,122131
3627 6364 7325
74 43
117 31
6364 5476
17316 2666
0,122132
7755 8428 11453
98 43
141 55
8428 9604
27636 4730 #p3a
0,122133
4345 1608 4633
67 12
79 55
1608 4489
10586 1320
0,1222
697 696 985
29 12 #Pell(5,4)
41 17
696 841
2378 408
0,12221
1275 2668 2957
46 29
75 17
2668 2116
6900 986 #p2a
0,122211
1853 5796 6085
63 46
109 17
5796 3969
13734 1564
0,122212
12525 11132 16757
121 46
167 75
11132 14641
40414 6900
0,122213
9975 6032 11657
104 29
133 75
6032 10816
27664 4350
0,12222
4059 4060 5741
70 29 #Pell(6,5)
99 41
4060 4900
13860 2378
0,122221
7421 15540 17221
111 70
181 41
15540 12321
40182 5740
0,122222
23661 23660 33461
169 70 #Pell(7,6)
239 99
23660 28561
80782 13860
0,122223
15543 7424 17225
128 29
157 99
7424 16384
40192 5742
0,12223
2665 1272 2953
53 12
65 41
1272 2809
6890 984
0,122231
6027 9964 11645
94 53
147 41
9964 8836
27636 4346 #p3a
0,122232
11115 12508 16733
118 53
171 65
12508 13924
40356 6890
0,122233
5785 1848 6073
77 12
89 65
1848 5929
13706 1560
0,1223
459 220 509
22 5
27 17
220 484
1188 170
0,12231
1037 1716 2005
39 22
61 17
1716 1521
4758 748
0,122311
1615 4368 4657
56 39
95 17
4368 3136
10640 1326
0,122312
8479 7800 11521
100 39
139 61
7800 10000
27800 4758
0,122313
6405 3652 7373
83 22
105 61
3652 6889
17430 2684
0,12232
1917 2156 2885
49 22
71 27
2156 2401
6958 1188
0,122321
3375 7448 8177
76 49
125 27
7448 5776
19000 2646
0,122322
11999 11760 16801
120 49
169 71
11760 14400
40560 6958
0,122323
8165 4092 9133
93 22
115 71
4092 8649
21390 3124
0,12233
999 320 1049
32 5
37 27
320 1024
2368 270
0,122331
2457 3776 4505
59 32
91 27
3776 3481
10738 1728
0,122332
3737 4416 5785
69 32
101 37
4416 4761
13938 2368
0,122333
1739 420 1789
42 5
47 37
420 1764
3948 370
0,123
77 36 85
9 2
11 7
36 81 not coprime
198 28
0,1231
1,323
175 288 337
16 9
25 7
288 256
800 126
0,12311
273 736 785
23 16
39 7
736 529
1794 224
0,123111
371 1380 1429
30 23
53 7
1380 900
3180 322
0,123112
3315 2852 4373
62 23
85 39
2852 3844
10540 1794
0,123113
2769 1760 3281
55 16
71 39
1760 3025
7810 1248
0,12312
1425 1312 1937
41 16
57 25
1312 1681
4674 800
0,123121
2675 5412 6037
66 41
107 25
5412 4356
14124 2050
0,123122
7923 8036 11285
98 41
139 57
8036 9604
27244 4674
0,123123
5073 2336 5585
73 16
89 57
2336 5329
12994 1824
0,12313
1075 612 1237
34 9
43 25
612 1156
2924 450
0,123131
2325 4012 4637
59 34
93 25
4012 3481
10974 1700
0,123132
4773 5236 7085
77 34
111 43
5236 5929
17094 2924
0,123133
2623 936 2785
52 9
61 43
936 2704
6344 774
0,1232
319 360 481
20 9
29 11
360 400
1160 198
0,12321
561 1240 1361
31 20
51 11
1240 961
3162 440
0,123211
803 2604 2725
42 31
73 11
2604 1764
6132 682
0,123212
5763 5084 7685
82 31
113 51
5084 6724
18532 3162
0,123213
4641 2840 5441
71 20
91 51
2840 5041
12922 2040
0,12322
2001 1960 2801
49 20
69 29
1960 2401
6762 1160
0,123221
3683 7644 8485
78 49
127 29
7644 6084
19812 2842
0,123222
11523 11564 16325
118 49
167 69
11564 13924
39412 6762
0,123223
7521 3560 8321
89 20
109 69
3560 7921
19402 2760
0,12323
1363 684 1525
38 9
47 29
684 1444
3572 522
0,123231
3045 5092 5933
67 38
105 29
5092 4489
14070 2204
0,123232
5781 6460 8669
85 38
123 47
6460 7225
20910 3572
0,123233
3055 1008 3217
56 9
65 47
1008 3136
7280 846
0,1233
1,1321
165 52 173
13 2
15 11
52 169
390 44
0,12331
407 624 745
24 13
37 11
624 576
1776 286
0,123311
649 1680 1801
35 24
59 11
1680 1225
4130 528
0,123312
3145 2928 4297
61 24
85 37
2928 3721
10370 1776
0,123313
2331 1300 2669
50 13
63 37
1300 2500
6300 962
0,12332
615 728 953
28 13
41 15
728 784
2296 390
0,123321
1065 2408 2633
43 28
71 15
2408 1849
6106 840
0,123322
3977 3864 5545
69 28
97 41
3864 4761
13386 2296
0,123323
2747 1404 3085
54 13
67 41
1404 2916
7236 1066
0,12333
285 68 293
17 2
19 15
68 289
646 60
0,123331
735 1088 1313
32 17
49 15
1088 1024
3136 510
0,123332
1,11122
1007 1224 1585
36 17
53 19
1224 1296
3816 646
0,123333
437 84 445
21 2
23 19
84 441n
966 76
0,13
1,2
15 8 17
4 1 #Pythagoras(2)
5 3
8 16
40 6
0,131
33 56 65
7 4
11 3
56 49
54 24
0,1311
51 140 149
10 7
17 3
140 100
340 42
0,13111
69 260 269
13 10
23 3
260 169
598 60
0,131111
87 416 425
16 13
29 3
416 256
928 78
0,131112
1127 936 1465
36 13
49 23
936 1296
3528 598
0,131113
989 660 1189
33 10
43 23
660 1089
2838 460
0,13112
629 540 829
27 10
37 17
540 729
1998 340
0,131121
1,22123
1207 2376 2665
44 27
71 17
2376 1936
6248 918
0,131122
3367 3456 4825
64 27
91 37
3456 4096
11648 1998
0,131123
2109 940 2309
47 10
57 37
940 2209
5358 740
0,13113
527 336 625
24 7
31 17
336 576
1488 238
0,131131
1105 1968 2257
41 24
65 17
1968 1681
5330 816
0,131132
2449 2640 3601
55 24
79 31
2640 3025
8690 1488
0,131133
1395 532 1493
38 7
45 31
532 1444
3420 434
0,1312
275 252 373
18 7
25 11
252 324
900 154
0,13121
517 1044 1165
29 18
47 11
1044 841
2726 396
0,131211
759 2320 2441
40 29
69 11
2320 1600
5520 638
0,131212
4935 4408 6617
76 29
105 47
4408 5776
15960 2726
0,131213
3901 2340 4549
65 18
83 47
2340 4225
10790 1692
0,13122
1525 1548 2173
43 18
61 25
1548 1849
5246 900
0,131221
2775 5848 6473
68 43
111 25
5848 4624
15096 2150
0,131222
8967 8944 12665
104 43
147 61
8944 10816
30576 5246
0,131223
5917 2844 6565
79 18
97 61
2844 6241
15326 2196
0,13123
975 448 1073
32 7
39 25
448 1024
2496 350
0,131231
2225 3648 4273
57 32
89 25
3648 3249
10146 1600
0,131232
4017 4544 6065
71 32
103 39
4544 5041
14626 2496
0,131233
2067 644 2165
46 7
53 39
644 2116
4876 546
0,1313
1,13211
209 120 241
15 4
19 11
120 225
570 88
0,13131
451 780 901
26 15
41 11
780 676
2132 330
0,131311
693 1924 2045
37 26
63 11
1924 1369
4662 572
0,131312
3813 3484 5165
67 26
93 41
3484 4489
12462 2132
0,131313
2911 1680 3361
56 15
71 41
1680 3136
7952 1230
0,13132
931 1020 1381
34 15
49 19
1020 1156
3332 570
0,131321
1653 3604 3965
53 34
87 19
3604 2809
9222 1292
0,131322
5733 5644 8045
83 34
117 49
5644 6889
19422 3332
0,131323
3871 1920 4321
64 15
79 49
1920 4096
10112 1470
0,13133
513 184 545
23 4
27 19
184 529
1242 152
0,131331
1235 1932 2293
42 23
65 19
1932 1764
5460 874
0,131332
1971 2300 3029
50 23
73 27
2300 2500
7300 1242
0,131333
945 248 977
31 4
35 27
248 961
2170 216
0,132
65 72 97
9 4
13 5
72 81
234 40
0,1321
1,1213
115 252 277
14 9
23 5
252 196
644 90
0,13211
1,1231
165 532 557
19 14
33 5
532 361
1254 140
0,132111
1,1233
215 912 937
24 19
43 5
912 576
2064 190
0,132112
2343 1976 3065
52 19
71 33
1976 2704
7384 1254
0,132113
2013 1316 2405
47 14
61 33
1316 2209
5734 924
0,13212
1173 1036 1565
37 14
51 23
1036 1369
3774 644
0,132121
2231 4440 4969
60 37
97 23
4440 3600
11640 1702
0,132122
6375 6512 9113
88 37
125 51
6512 7744
22000 3774
0,132123
4029 1820 4421
65 14
79 51
1820 4225
10270 1428
0,13213
943 576 1105
32 9
41 23
576 1024
2624 414
0,132131
2001 3520 4049
55 32
87 23
3520 3025
9570 1472
0,132132
4305 4672 6353
73 32
105 41
4672 5329
15330 2624
0,132133
2419 900 2581
50 9
59 41
900 2500
5900 738
0,1322
403 396 565
22 9
31 13
396 484
1364 234
0,13221
741 1540 1709
35 22
57 13
1540 1225
3990 572
0,132211
1,13223
1079 3360 3529
48 35
83 13
3360 2304
7968 910
0,132212
7239 6440 9689
92 35
127 57
6440 8464
23368 3990
0,132213
5757 3476 6725
79 22
101nbsp;57
3476 6241
15958 2508
0,13222
2325 2332 3293
53 22
75 31
2332 2809
7950 1364
0,132221
4247 8904 9865
84 53
137 31
8904 7056
23016 3286
0,132222
13575 13568 19193
128 53
181 75
13568 16384
46336 7950
0,132223
8925 4268 9893
97 22
119 75
4268 9409
23086 3300
0,13223
1,11232
1519 720 1681
40 9
49 31
720 1600
3920 558
0,132231
3441 5680 6641
71 40
111 31
5680 5041
15762 2480
0,132232
6321 7120 9521
89 40
129 49
7120 7921
22962 3920
0,132233
3283 1044 3445
58 9
67 49
1044 3364
7772 882
0,1323
273 136 305
17 4
21 13
136 289
714 104
0,13231
611 1020 1189
30 17
47 13
1020 900
2820 442
0,132311
949 2580 2749
43 30
73 13
2580 1849
6278 780
0,132312
5029 4620 6829
77 30
107 47
4620 5929
16478 2820
0,132313
3807 2176 4385
64 17
81 47
2176 4096
10368 1598
0,13232
1155 1292 1733
38 17
55 21
1292 1444
4180 714
0,132321
2037 4484 4925
59 38
97 21
4484 3481
11446 1596
0,132322
7205 7068 10093
93 38
131 55
7068 8649
24366 4180
0,132323
4895 2448 5473
72 17
89 55
2448 5184
12816 1870
0,13233
609 200 641
25 4
29 21
200 625
1450 168
0,132331
1491 2300 2741
46 25
71 21
2300 2116
6532 1050
0,132332
2291 2700 3541
54 25
79 29
2700 2916
8532 1450
0,132333
1073 264 1105
33 4
37 29
264 1089
2442 232
0,133
35 12 37
6 1 #Pythagoras(3)
7 5
12 36
84 10
0,1331
85 132 157
11 6
17 5
132 121
374 60
0,13311
1,223
135 352 377
16 11
27 5
352 256
864 110
0,133111
185 672 697
21 16
37 5
672 441
1554 160
0,133112
1593 1376 2105
43 16
59 27
1376 1849
5074 864
0,133113
1323 836 1565
38 11
49 27
836 1444
3724 594
0,13312
663 616 905
28 11
39 17
616 784
2184 374
0,133121
1241 2520 2809
45 28
73 17
2520 2025
6570 952
0,133122
3705 3752 5273
67 28
95 39
3752 4489
12730 2184
0,133123
2379 1100 2621
50 11
61 39
1100 2500
6100 858
0,13313
493 276 565
23 6
b n
29 17
276 529
1334 204
0,133131
1,21323
1071 1840 2129
40 23
63 17
1840 1600
5040 782
0,133132
1,23122
2175 2392 3233
52 23
75 29
2392 2704
7800 1334
0,133133
1,23321
1189 420 1261
35 6
41 29
420 1225
2870 348
0,1332
1,2121
133 156 205
13 6
19 7
156 169
494 84
0,13321
231 520 569
20 13
33 7
520 400
1320 182
0,133211
329 1080 1129
27 20
47 7
1080 729
2538 280
0,133212
2409 2120 3209
53 20
73 33
2120 2809
7738 1320
0,133213
1947 1196 2285
46 13
59 33
1196 2116
5428 858
0,13322
855 832 1193
32 13
45 19
832 1024
2880 494
0,133221
1577 3264 3625
51 32
83 19
3264 2601
8466 1216
0,133222
4905 4928 6953
77 32
109 45
4928 5929
16786 2880
0,133223
3195 1508 3533
58 13
71 45
1508 3364
8236 1170
0,13323
589 300 661
25 6
31 19
300 625
1550 228
0,133231
1311 2200 2561
44 25
69 19
2200 1936
6072 950
0,133232
2511 2800 3761
56 25
81 31
2800 3136
9072 1550
0,133233
1333 444 1405
37 6
43 31
444 1369
3182 372
0,1333
63 16 65
8 1 #Pythagoras(4)
9 7
16 64
144 14
0,13331
1,32111
161 240 289
15 8
23 7
240 225
690 112
0,133311
259 660 709
22 15
37 7
660 484
1628 210
0,133312
1219 1140 1669
38 15
53 23
1140 1444
4028 690
0,133313
897 496 1025
31 8
39 23
496 961
2418 368
0,13332
225 272 353
17 8
25 9
272 289
850 144
0,133321
387 884 965
26 17
43 9
884 676
2236 306
0,133322
1475 1428 2053
42 17
59 25
1428 1764
4956 850
0,133323
1025 528 1153
33 8
41 25
528 1089
2706 400
0,13333
99 20 101
10 1 #Pythagoras(5)
11 9
20 100
220 18
0,133331
261 380 461
19 10
29 9
380 361
1102 180
0,133332
341 420 541
21 10
31 11
420 441
1302 220
0,133333
1,132
143 24 145
12 1 #Pythagoras(6)
13 11
24 144
312 22
top of table


Notes: added some (u, v) pairs ; intermittent saving regards Gangleri 16:32, 13 July 2019 (UTC)

details to Berggrens's tree[edit]

to be continued 2A02:810D:41C0:134:6C5D:2ACF:DA6:D533 (talk) 18:07, 12 July 2019 (UTC)

sorted (b, n) pairs for the Berggrens's tree regards

4 4 12 9 24 16 40 25 60 36 84 49 204 289 160 256 140 196 644 529 924 1089 240 576 104 169 572 484 780 900 168 441 88 121 396 324 900 625 1692 2209 880 1600 572 676 2132 1681 3276 3969 1056 2304 152 361 1292 1156 1596 1764 216 729 60 100 340 289 816 576 1496 1936 740 1369 460 529 1656 1296 2576 3136 860 1849 96 256 928 841 1120 1225 132 484 48 64 208 169 468 324 828 529 1764 2401 1144 1936 884 1156 3740 3025 5508 6561 1560 3600 464 841 2900 2500 3828 4356 720 2025 304 361 1140 900 2460 1681 4740 6241 2584 4624 1748 2116 6716 5329 10212 12321 3192 7056 560 1225 4340 3844 5460 6084 816 2601 84 196 700 625 1800 1296 3200 4096 1484 2809 868 961 2976 2304 4712 5776 1652 3481 120 400 1480 1369 1720 1849 156 676 28 49 168 144 408 289 748 484 1564 2116 984 1681 744 961 3100 2500 4588 5476 1320 3025 364 676 2340 2025 3068 3481 560 1600 224 256 800 625 1700 1156 3300 4356 1824 3249 1248 1521 4836 3844 7332 8836 2272 5041 420 900 3180 2809 4020 4489 616 1936 44 121 440 400 1160 841 2040 2601 924 1764 528 576 1776 1369 2832 3481 1012 2116 60 225 840 784 960 1024 76 361 20 25 80 64 176 121 308 196 476 289 1092 1521 792 1296 660 900 2940 2401 4260 5041 1144 2704 432 729 2484 2116 3348 3844 688 1849 336 441 1428 1156 3196 2209 6052 7921 3192 5776 2100 2500 7900 6241 12100 14641 3864 8464 592 1369 4884 4356 6068 6724 848 2809 180 324 1116 961 2728 1936 4960 6400 2412 4489 1476 1681 5248 4096 8200 10000 2772 5929 280 784 2856 2601 3416 3721 380 1444 120 144 456 361 988 676 1716 1089 3692 5041 2432 4096 1900 2500 8100 6561 11900 14161 3344 7744 1032 1849 6364 5476 8428 9604 1608 4489 696 841 2668 2116 5796 3969 11132 14641 6032 10816 4060 4900 15540 12321 23660 28561 7424 16384 1272 2809 9964 8836 12508 13924 1848 5929 220 484 1716 1521 4368 3136 7800 10000 3652 6889 2156 2401 7448 5776 11760 14400 4092 8649 320 1024 3776 3481 4416 4761 420 1764 36 81 288 256 736 529 1380 900 2852 3844 1760 3025 1312 1681 5412 4356 8036 9604 2336 5329 612 1156 4012 3481 5236 5929 936 2704 360 400 1240 961 2604 1764 5084 6724 2840 5041 1960 2401 7644 6084 11564 13924 3560 7921 684 1444 5092 4489 6460 7225 1008 3136 52 169 624 576 1680 1225 2928 3721 1300 2500 728 784 2408 1849 3864 4761 1404 2916 68 289 1088 1024 1224 1296 84 441 8 16 56 49 140 100 260 169 416 256 936 1296 660 1089 540 729 2376 1936 3456 4096 940 2209 336 576 1968 1681 2640 3025 532 1444 252 324 1044 841 2320 1600 4408 5776 2340 4225 1548 1849 5848 4624 8944 10816 2844 6241 448 1024 3648 3249 4544 5041 644 2116 120 225 780 676 1924 1369 3484 4489 1680 3136 1020 1156 3604 2809 5644 6889 1920 4096 184 529 1932 1764 2300 2500 248 961 72 81 252 196 532 361 912 576 1976 2704 1316 2209 1036 1369 4440 3600 6512 7744 1820 4225 576 1024 3520 3025 4672 5329 900 2500 396 484 1540 1225 3360 2304 6440 8464 3476 6241 2332 2809 8904 7056 13568 16384 4268 9409 720 1600 5680 5041 7120 7921 1044 3364 136 289 1020 900 2580 1849 4620 5929 2176 4096 1292 1444 4484 3481 7068 8649 2448 5184 200 625 2300 2116 2700 2916 264 1089 12 36 132 121 352 256 672 441 1376 1849 836 1444 616 784 2520 2025 3752 4489 1100 2500 276 529 1840 1600 2392 2704 420 1225 156 169 520 400 1080 729 2120 2809 1196 2116 832 1024 3264 2601 4928 5929 1508 3364 300 625 2200 1936 2800 3136 444 1369 16 64 240 225 660 484 1140 1444 496 961 272 289 884 676 1428 1764 528 1089 20 100 380 361 420 441 24 144

added sorted (b, n) pairs for the Berggrens's tree; regards Gangleri 217.86.249.144 (talk) 13:14, 16 July 2019 (UTC)

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triplets contained in both trees

1_0_1|0 1 0 1 1_0_1|0 1 0 1 1 1005_2132_2357|0,111221 1005 2132 2357 1005_2132_2357|1,31221 1005 2132 2357 1 1007_1224_1585|0,123332 1007 1224 1585 1007_1224_1585|1,11122 1007 1224 1585 1 101_5100_5101|1,31131 101 5100 5101 1015_192_1033|1,2332 1015 192 1033 10205_10212_14437|0,112222 10205 10212 14437 1023_64_1025|1,3332 1023 64 1025 1025_528_1153|0,133323 1025 528 1153 103_5304_5305|1,31133 103 5304 5305 1037_1716_2005|0,12231 1037 1716 2005 1045_4452_4573|1,22231 1045 4452 4573 105_208_233|0,1121 105 208 233 105_208_233|1,12111 105 208 233 1 105_5512_5513|1,31311 105 5512 5513 105_608_617|1,21111 105 608 617 105_88_137|0,1112 105 88 137 105_88_137|1,3211 105 88 137 1 1053_3196_3365|0,121211 1053 3196 3365 1065_2408_2633|0,123321 1065 2408 2633 107_5724_5725|1,31313 107 5724 5725 1071_1840_2129|0,133131 1071 1840 2129 1071_1840_2129|1,21323 1071 1840 2129 1 1071_7040_7121|1,32233 1071 7040 7121 1073_264_1105|0,132333 1073 264 1105 1075_612_1237|0,12313 1075 612 1237 1079_3360_3529|0,132211 1079 3360 3529 1079_3360_3529|1,13223 1079 3360 3529 1 1085_132_1093|1,33321 1085 132 1093 109_5940_5941|1,31331 109 5940 5941 11_60_61|0,11111 11 60 61 11_60_61|1,13 11 60 61 1 1105_1968_2257|0,131131 1105 1968 2257 1107_476_1205|1,12212 1107 476 1205 111_6160_6161|1,31333 111 6160 6161 111_680_689|1,2133 111 680 689 11115_12508_16733|0,122232 11115 12508 16733 1127_936_1465|0,131112 1127 936 1465 113_6384_6385|1,33111 113 6384 6385 1131_340_1181|1,22112 1131 340 1181 1147_204_1165|1,21312 1147 204 1165 115_252_277|0,1321 115 252 277 115_252_277|1,1213 115 252 277 1 115_6612_6613|1,33113 115 6612 6613 11523_11564_16325|0,123222 11523 11564 16325 1155_1292_1733|0,13232 1155 1292 1733 1155_68_1157|1,11112 1155 68 1157 1157_3876_4045|1,23231 1157 3876 4045 1159_1680_2041|1,11323 1159 1680 2041 1161_560_1289|0,11223 1161 560 1289 11661_11900_16661|0,122122 11661 11900 16661 117_44_125|0,1133 117 44 125 117_44_125|1,1121 117 44 125 1 117_6844_6845|1,33131 117 6844 6845 1173_1036_1565|0,13212 1173 1036 1565 1175_792_1417|0,121113 1175 792 1417 1175_792_1417|1,32122 1175 792 1417 1 1189_420_1261|0,133133 1189 420 1261 1189_420_1261|1,23321 1189 420 1261 1 119_120_169|0,122 119 120 169 119_120_169|1,122 119 120 169 1 119_7080_7081|1,33133 119 7080 7081 11999_11760_16801|0,122322 11999 11760 16801 1207_2376_2665|0,131121 1207 2376 2665 1207_2376_2665|1,22123 1207 2376 2665 1 121_7320_7321|1,33311 121 7320 7321 12141_12100_17141|0,121222 12141 12100 17141 1219_1140_1669|0,133312 1219 1140 1669 123_7564_7565|1,33313 123 7564 7565 123_836_845|1,21113 123 836 845 1235_1932_2293|0,131331 1235 1932 2293 1239_1520_1961|1,11322 1239 1520 1961 1241_2520_2809|0,133121 1241 2520 2809 1245_3332_3557|1,33231 1245 3332 3557 1247_504_1345|1,32132 1247 504 1345 125_7812_7813|1,33331 125 7812 7813 12525_11132_16757|0,122212 12525 11132 16757 1265_1248_1777|0,11322 1265 1248 1777 1269_740_1469|0,111313 1269 740 1469 1269_740_1469|1,22321 1269 740 1469 1 127_8064_8065|1,33333 127 8064 8065 1275_2668_2957|0,12221 1275 2668 2957 1275_988_1613|1,23212 1275 988 1613 1287_4816_4985|1,31233 1287 4816 4985 1287_6784_6905|1,22233 1287 6784 6905 129_920_929|1,2311 129 920 929 1295_72_1297|1,11132 1295 72 1297 13_84_85|0,111111 13 84 85 13_84_85|1,31 13 84 85 1 1305_592_1433|0,12123 1305 592 1433 1309_2820_3109|1,33221 1309 2820 3109 1311_1360_1889|1,21322 1311 1360 1889 1311_2200_2561|0,133231 1311 2200 2561 1323_836_1565|0,133113 1323 836 1565 1325_1092_1717|0,121112 1325 1092 1717 1325_1092_1717|1,32321 1325 1092 1717 1 133_156_205|0,1332 133 156 205 133_156_205|1,2121 133 156 205 1 1333_444_1405|0,133233 1333 444 1405 1343_2976_3265|0,112321 1343 2976 3265 1343_2976_3265|1,12223 1343 2976 3265 1 1349_2340_2701|0,113131 1349 2340 2701 1349_2340_2701|1,23221 1349 2340 2701 1 135_352_377|0,13311 135 352 377 135_352_377|1,223 135 352 377 1 1357_1476_2005|0,12132 1357 1476 2005 1357_1476_2005|1,32221 1357 1476 2005 1 13575_13568_19193|0,132222 13575 13568 19193 1363_684_1525|0,12323 1363 684 1525 1365_1892_2333|1,22221 1365 1892 2333 1387_2484_2845|0,121131 1387 2484 2845 1395_532_1493|0,131133 1395 532 1493 1403_1596_2125|0,111232 1403 1596 2125 1407_2024_2465|1,13123 1407 2024 2465 141_1100_1109|1,21131 141 1100 1109 1419_380_1469|0,121333 1419 380 1469 1419_380_1469|1,12312 1419 380 1469 1 1425_1312_1937|0,12312 1425 1312 1937 143_24_145|0,133333 143 24 145 143_24_145|1,132 143 24 145 1 1431_1040_1769|1,21222 1431 1040 1769 1443_76_1445|1,11312 1443 76 1445 1449_1720_2249|0,112332 1449 1720 2249 145_408_433|0,11311 145 408 433 145_408_433|1,2211 145 408 433 1 1455_4592_4817|1,31223 1455 4592 4817 1463_2784_3145|1,12323 1463 2784 3145 147_1196_1205|1,2313 147 1196 1205 1475_1428_2053|0,133322 1475 1428 2053 1479_880_1721|0,111213 1479 880 1721 1479_880_1721|1,11222 1479 880 1721 1 1491_2300_2741|0,132331 1491 2300 2741 1495_1848_2377|1,13122 1495 1848 2377 1495_6528_6697|1,23233 1495 6528 6697 15_112_113|1,33 15 112 113 15_8_17|0,13 15 8 17 15_8_17|1,2 15 8 17 1 1501_2940_3301|0,121121 1501 2940 3301 1519_720_1681|0,13223 1519 720 1681 1519_720_1681|1,11232 1519 720 1681 1 1525_1548_2173|0,13122 1525 1548 2173 153_104_185|0,11113 153 104 185 153_104_185|1,11211 153 104 185 1 1537_984_1825|0,113113 1537 984 1825 1539_3100_3461|0,113121 1539 3100 3461 155_468_493|0,11211 155 468 493 155_468_493|1,12113 155 468 493 1 1551_560_1649|0,113133 1551 560 1649 1551_560_1649|1,21232 1551 560 1649 1 15543_7424_17225|0,122223 15543 7424 17225 1577_3264_3625|0,133221 1577 3264 3625 159_1400_1409|1,21133 159 1400 1409 1591_240_1609|1,21332 1591 240 1609 1593_1376_2105|0,133112 1593 1376 2105 1599_80_1601|1,11332 1599 80 1601 161_240_289|0,13331 161 240 289 161_240_289|1,32111 161 240 289 1 1615_4368_4657|0,122311 1615 4368 4657 1643_924_1885|0,113313 1643 924 1885 1643_924_1885|1,22212 1643 924 1885 1 1647_1496_2225|0,111312 1647 1496 2225 1647_1496_2225|1,22122 1647 1496 2225 1 165_1508_1517|1,2331 165 1508 1517 165_52_173|0,1233 165 52 173 165_52_173|1,1321 165 52 173 1 165_532_557|0,13211 165 532 557 165_532_557|1,1231 165 532 557 1 1653_3604_3965|0,131321 1653 3604 3965 1659_2900_3341|0,112131 1659 2900 3341 1679_2400_2929|1,13323 1679 2400 2929 1695_6272_6497|1,33233 1695 6272 6497 17_144_145|1,111 17 144 145 1705_1032_1993|0,12213 1705 1032 1993 171_140_221|0,11112 171 140 221 171_140_221|1,1212 171 140 221 1 1739_420_1789|0,122333 1739 420 1789 1749_860_1949|0,111323 1749 860 1949 175_288_337|0,1231 175 288 337 175_288_337|1,323 175 288 337 1 1755_1748_2477|0,11222 1755 1748 2477 1763_84_1765|1,13112 1763 84 1765 1767_1144_2105|0,112113 1767 1144 2105 177_1736_1745|1,21311 177 1736 1745 1775_2208_2833|1,13322 1775 2208 2833 1785_688_1913|0,121133 1785 688 1913 1817_2856_3385|0,121331 1817 2856 3385 1827_1564_2405|0,113112 1827 1564 2405 183_1856_1865|1,2333 183 1856 1865 185_672_697|0,133111 185 672 697 1853_5796_6085|0,122211 1853 5796 6085 1863_3016_3545|1,23123 1863 3016 3545 1869_3740_4181|0,112121 1869 3740 4181 187_84_205|0,1123 187 84 205 187_84_205|1,2112 187 84 205 1 1885_1692_2533|0,111212 1885 1692 2533 1887_6016_6305|1,33223 1887 6016 6305 1887_616_1985|0,113233 1887 616 1985 189_340_389|0,11131 189 340 389 189_340_389|1,1221 189 340 389 1 19_180_181|1,113 19 180 181 1909_3180_3709|0,113231 1909 3180 3709 1911_440_1961|1,22132 1911 440 1961 1917_2156_2885|0,12232 1917 2156 2885 1935_88_1937|1,13132 1935 88 1937 1943_1824_2665|1,12322 1943 1824 2665 1947_1196_2285|0,133213 1947 1196 2285 195_2108_2117|1,21313 195 2108 2117 195_28_197|1,312 195 28 197 195_748_773|0,113111 195 748 773 195_748_773|1,2213 195 748 773 1 1961_720_2089|0,112133 1961 720 2089 1971_2300_3029|0,131332 1971 2300 3029 1975_2808_3433|1,31123 1975 2808 3433 1995_1012_2237|0,113323 1995 1012 2237 2001_1960_2801|0,12322 2001 1960 2801 2001_3520_4049|0,132131 2001 3520 4049 2013_1316_2405|0,132113 2013 1316 2405 2015_1632_2593|1,12222 2015 1632 2593 203_396_445|0,11121 203 396 445 203_396_445|1,3213 203 396 445 1 2035_828_2197|1,32212 2035 828 2197 2037_4484_4925|0,132321 2037 4484 4925 205_828_853|0,112111 205 828 853 205_828_853|1,12131 205 828 853 1 2059_2100_2941|0,12122 2059 2100 2941 2067_644_2165|0,131233 2067 644 2165 207_224_305|0,1132 207 224 305 207_224_305|1,322 207 224 305 1 2071_5760_6121|1,23223 2071 5760 6121 2077_1764_2725|0,112112 2077 1764 2725 2079_2600_3329|1,31122 2079 2600 3329 209_120_241|0,1313 209 120 241 209_120_241|1,13211 209 120 241 1 21_20_29|0,12 21 20 29 21_20_29|1,21 21 20 29 1 21_220_221|1,131 21 220 221 2107_276_2125|1,23112 2107 276 2125 2109_940_2309|0,131123 2109 940 2309 2115_92_2117|1,13312 2115 92 2117 213_2516_2525|1,21331 213 2516 2525 2135_1248_2473|1,13222 2135 1248 2473 2139_1900_2861|0,12212 2139 1900 2861 215_912_937|0,132111 215 912 937 215_912_937|1,1233 215 912 937 1 217_456_505|0,1221 217 456 505 217_456_505|1,21211 217 456 505 1 2175_2392_3233|0,133132 2175 2392 3233 2175_2392_3233|1,23122 2175 2392 3233 1 2183_1056_2425|0,111223 2183 1056 2425 2183_1056_2425|1,13232 2183 1056 2425 1 2201_2040_3001|0,113312 2201 2040 3001 221_60_229|0,11333 221 60 229 221_60_229|1,3121 221 60 229 1 2225_3648_4273|0,131231 2225 3648 4273 2231_4440_4969|0,132121 2231 4440 4969 2247_5504_5945|1,22223 2247 5504 5945 225_272_353|0,13332 225 272 353 2255_672_2353|1,12232 2255 672 2353 2279_480_2329|1,12332 2279 480 2329 2291_2700_3541|0,132332 2291 2700 3541 2295_3248_3977|1,31323 2295 3248 3977 23_264_265|1,133 23 264 265 2303_96_2305|1,13332 2303 96 2305 231_160_281|0,111113 231 160 281 231_160_281|1,222 231 160 281 1 231_2960_2969|1,21333 231 2960 2969 231_520_569|0,13321 231 520 569 231_520_569|1,2123 231 520 569 1 2323_4836_5365|0,113221 2323 4836 5365 2325_2332_3293|0,13222 2325 2332 3293 2325_4012_4637|0,123131 2325 4012 4637 2331_1300_2669|0,123313 2331 1300 2669 2343_1976_3065|0,132112 2343 1976 3065 23661_23660_33461|0,122222 23661 23660 33461 2379_1100_2621|0,133123 2379 1100 2621 2407_3024_3865|1,31322 2407 3024 3865 2409_2120_3209|0,133212 2409 2120 3209 2415_5248_5777|0,121321 2415 5248 5777 2415_5248_5777|1,32223 2415 5248 5777 1 2419_900_2581|0,132133 2419 900 2581 2449_2640_3601|0,131132 2449 2640 3601 245_1188_1213|1,2231 245 1188 1213 2451_700_2549|1,32312 2451 700 2549 2457_3776_4505|0,122331 2457 3776 4505 247_96_265|0,11133 247 96 265 247_96_265|1,232 247 96 265 1 249_3440_3449|1,23111 249 3440 3449 2499_100_2501|1,31112 2499 100 2501 25_312_313|1,311 25 312 313 2511_2800_3761|0,133232 2511 2800 3761 253_204_325|0,111112 253 204 325 253_204_325|1,21121 253 204 325 1 2537_816_2665|0,112233 2537 816 2665 255_1288_1313|1,12133 255 1288 1313 255_32_257|1,332 255 32 257 2575_4992_5617|1,32323 2575 4992 5617 2583_1144_2825|0,121123 2583 1144 2825 259_660_709|0,133311 259 660 709 259_660_709|1,32113 259 660 709 1 2607_2576_3665|0,111322 2607 2576 3665 261_380_461|0,133331 261 380 461 261_380_461|1,12121 261 380 461 1 2613_1484_3005|0,112313 2613 1484 3005 2619_4340_5069|0,112231 2619 4340 5069 2623_936_2785|0,123133 2623 936 2785 2639_3720_4561|1,33123 2639 3720 4561 265_1392_1417|1,22111 265 1392 1417 2665_1272_2953|0,12223 2665 1272 2953 267_3956_3965|1,23113 267 3956 3965 2675_5412_6037|0,123121 2675 5412 6037 2695_312_2713|1,23132 2695 312 2713 27_364_365|1,313 27 364 365 2703_104_2705|1,31132 2703 104 2705 2727_4736_5465|1,22323 2727 4736 5465 273_136_305|0,1323 273 136 305 273_136_305|1,31211 273 136 305 1 273_736_785|0,12311 273 736 785 2745_848_2873|0,121233 2745 848 2873 2747_1404_3085|0,123323 2747 1404 3085 275_252_373|0,1312 275 252 373 275_252_373|1,3212 275 252 373 1 2759_3480_4441|1,33122 2759 3480 4441 2769_1760_3281|0,123113 2769 1760 3281 2775_5848_6473|0,131221 2775 5848 6473 279_440_521|0,11331 279 440 521 279_440_521|1,1123 279 440 521 1 2805_3068_4157|0,113132 2805 3068 4157 285_4508_4517|1,23131 285 4508 4517 285_68_293|0,12333 285 68 293 285_68_293|1,3321 285 68 293 1 287_816_865|0,111311 287 816 865 287_816_865|1,1223 287 816 865 1 2871_4480_5321|1,23323 2871 4480 5321 2881_1320_3169|0,113123 2881 1320 3169 2891_540_2941|1,22312 2891 540 2941 29_420_421|1,331 29 420 421 2905_2832_4057|0,113322 2905 2832 4057 2911_1680_3361|0,131313 2911 1680 3361 2911_1680_3361|1,31222 2911 1680 3361 1 2915_108_2917|1,31312 2915 108 2917 2937_3416_4505|0,121332 2937 3416 4505 295_1728_1753|1,2233 295 1728 1753 2967_1456_3305|1,31232 2967 1456 3305 297_304_425|0,1122 297 304 425 2987_4884_5725|0,121231 2987 4884 5725 299_180_349|0,1213 299 180 349 299_180_349|1,12112 299 180 349 1 2993_1824_3505|0,113213 2993 1824 3505 3_4_5|0,1 3 4 5 3_4_5|1 3 4 5 1 3007_4224_5185|1,33323 3007 4224 5185 301_900_949|0,111211 301 900 949 301_900_949|1,3231 301 900 949 1 303_5096_5105|1,23133 303 5096 5105 3045_5092_5933|0,123231 3045 5092 5933 305_1848_1873|1,12311 305 1848 1873 3055_1008_3217|0,123233 3055 1008 3217 31_480_481|1,333 31 480 481 3115_3348_4573|0,121132 3115 3348 4573 3135_112_3137|1,31332 3135 112 3137 3135_3968_5057|1,33322 3135 3968 5057 3145_2928_4297|0,123312 3145 2928 4297 315_1972_1997|1,22113 315 1972 1997 315_572_653|0,111131 315 572 653 315_572_653|1,11213 315 572 653 1 315_988_1037|0,12211 315 988 1037 315_988_1037|1,21213 315 988 1037 1 319_360_481|0,1232 319 360 481 319_360_481|1,1122 319 360 481 1 3195_1508_3533|0,133223 3195 1508 3533 321_5720_5729|1,23311 321 5720 5729 3213_6716_7445|0,112221 3213 6716 7445 323_36_325|1,1112 323 36 325 325_228_397|1,2321 325 228 397 3255_3712_4937|1,23322 3255 3712 4937 3283_1044_3445|0,132233 3283 1044 3445 3285_1652_3677|0,112323 3285 1652 3677 329_1080_1129|0,133211 329 1080 1129 329_1080_1129|1,12211 329 1080 1129 1 3293_3276_4645|0,111222 3293 3276 4645 33_544_545|1,1111 33 544 545 33_56_65|0,131 33 56 65 33_56_65|1,211 33 56 65 1 3315_2852_4373|0,123112 3315 2852 4373 333_644_725|0,111121 333 644 725 333_644_725|1,3221 333 644 725 1 3355_348_3373|1,23312 3355 348 3373 3363_116_3365|1,33112 3363 116 3365 3367_3456_4825|0,131122 3367 3456 4825 3367_3456_4825|1,22322 3367 3456 4825 1 3375_7448_8177|0,122321 3375 7448 8177 339_6380_6389|1,23313 339 6380 6389 341_420_541|0,133332 341 420 541 341_420_541|1,2221 341 420 541 1 3431_1560_3769|0,112123 3431 1560 3769 3441_5680_6641|0,132231 3441 5680 6641 345_152_377|0,11123 345 152 377 345_152_377|1,33211 345 152 377 1 3471_3200_4721|0,112312 3471 3200 4721 3471_3200_4721|1,32322 3471 3200 4721 1 35_12_37|0,133 35 12 37 35_12_37|1,12 35 12 37 1 35_612_613|1,1113 35 612 613 351_280_449|1,2122 351 280 449 3515_3828_5197|0,112132 3515 3828 5197 355_2508_2533|1,12313 355 2508 2533 3567_2944_4625|1,32222 3567 2944 4625 357_1276_1325|1,32131 357 1276 1325 357_7076_7085|1,23331 357 7076 7085 357_76_365|0,113333 357 76 365 357_76_365|1,11121 357 76 365 1 3589_4020_5389|0,113232 3589 4020 5389 3599_120_3601|1,33132 3599 120 3601 3627_6364_7325|0,122131 3627 6364 7325 365_2652_2677|1,22131 365 2652 2677 3655_2688_4537|1,22222 3655 2688 4537 3683_7644_8485|0,123221 3683 7644 8485 369_800_881|0,11321 369 800 881 37_684_685|1,1131 37 684 685 3705_3752_5273|0,133122 3705 3752 5273 371_1380_1429|0,123111 371 1380 1429 3731_3300_4981|0,113212 3731 3300 4981 3735_2432_4457|0,122113 3735 2432 4457 3735_2432_4457|1,23222 3735 2432 4457 1 3737_4416_5785|0,122332 3737 4416 5785 3741_7900_8741|0,121221 3741 7900 8741 375_7808_7817|1,23333 375 7808 7817 377_336_505|0,1212 377 336 505 3807_2176_4385|0,132313 3807 2176 4385 3807_2176_4385|1,33222 3807 2176 4385 1 3813_3484_5165|0,131312 3813 3484 5165 3843_124_3845|1,33312 3843 124 3845 387_884_965|0,133321 387 884 965 3871_1920_4321|0,131323 3871 1920 4321 3871_1920_4321|1,33232 3871 1920 4321 1 39_760_761|1,1133 39 760 761 39_80_89|0,121 39 80 89 39_80_89|1,23 39 80 89 1 3901_2340_4549|0,131213 3901 2340 4549 391_120_409|0,11233 391 120 409 391_120_409|1,2132 391 120 409 1 3927_1664_4265|1,23232 3927 1664 4265 3975_1408_4217|1,22232 3975 1408 4217 3977_3864_5545|0,123322 3977 3864 5545 399_1600_1649|1,3233 399 1600 1649 399_40_401|1,1132 399 40 401 4015_1152_4177|1,32232 4015 1152 4177 4017_4544_6065|0,131232 4017 4544 6065 4029_1820_4421|0,132123 4029 1820 4421 403_396_565|0,1322 403 396 565 403_396_565|1,11212 403 396 565 1 4047_896_4145|1,32332 4047 896 4145 405_3268_3293|1,12331 405 3268 3293 4059_4060_5741|0,12222 4059 4060 5741 4061_8100_9061|0,122121 4061 8100 9061 407_624_745|0,12331 407 624 745 407_624_745|1,1323 407 624 745 1 4071_640_4121|1,22332 4071 640 4121 4087_384_4105|1,23332 4087 384 4105 4095_128_4097|1,33332 4095 128 4097 41_840_841|1,1311 41 840 841 413_1716_1765|0,122111 413 1716 1765 413_1716_1765|1,21231 413 1716 1765 1 4141_4260_5941|0,121122 4141 4260 5941 415_3432_3457|1,22133 415 3432 3457 4165_2412_4813|0,121313 4165 2412 4813 423_1064_1145|1,12123 423 1064 1145 4247_8904_9865|0,132221 4247 8904 9865 425_168_457|0,111133 425 168 457 4263_2584_4985|0,112213 4263 2584 4985 427_1836_1885|1,12213 427 1836 1885 429_460_629|0,11132 429 460 629 429_700_821|0,11231 429 700 821 429_700_821|1,32121 429 700 821 1 43_924_925|1,1313 43 924 925 4305_4672_6353|0,132132 4305 4672 6353 4345_1608_4633|0,122133 4345 1608 4633 435_308_533|1,32112 435 308 533 4365_3692_5717|0,122112 4365 3692 5717 437_84_445|0,123333 437 84 445 437_84_445|1,11321 437 84 445 1 441_1160_1241|0,113311 441 1160 1241 45_1012_1013|1,1331 45 1012 1013 45_28_53|0,113 45 28 53 45_28_53|1,121 45 28 53 1 451_780_901|0,13131 451 780 901 451_780_901|1,13213 451 780 901 1 4515_4588_6437|0,113122 4515 4588 6437 455_2088_2137|1,32133 455 2088 2137 455_4128_4153|1,12333 455 4128 4153 455_528_697|0,11332 455 528 697 455_528_697|1,1322 455 528 697 1 459_220_509|0,1223 459 220 509 459_220_509|1,2212 459 220 509 1 4641_2840_5441|0,123213 4641 2840 5441 465_4312_4337|1,22311 465 4312 4337 47_1104_1105|1,1333 47 1104 1105 475_132_493|0,111333 475 132 493 475_132_493|1,21112 475 132 493 1 477_1364_1445|1,11231 477 1364 1445 4773_5236_7085|0,123132 4773 5236 7085 4785_2272_5297|0,113223 4785 2272 5297 481_600_769|1,23211 481 600 769 4815_4712_6737|0,112322 4815 4712 6737 483_44_485|1,1312 483 44 485 4859_5460_7309|0,112232 4859 5460 7309 4895_2448_5473|0,132323 4895 2448 5473 49_1200_1201|1,3111 49 1200 1201 4905_4928_6953|0,133222 4905 4928 6953 493_276_565|0,13313 493 276 565 493_276_565|1,21321 493 276 565 1 4935_4408_6617|0,131212 4935 4408 6617 495_1472_1553|1,3223 495 1472 1553 495_952_1073|1,21123 495 952 1073 5_12_13|0,11 5 12 13 5_12_13|1,1 5 12 13 1 5029_4620_6829|0,132312 5029 4620 6829 5073_2336_5585|0,123123 5073 2336 5585 51_1300_1301|1,3113 51 1300 1301 51_140_149|0,1311 51 140 149 51_140_149|1,213 51 140 149 1 511_2640_2689|1,21233 511 2640 2689 513_184_545|0,13133 513 184 545 515_5292_5317|1,22313 515 5292 5317 517_1044_1165|0,13121 517 1044 1165 517_1044_1165|1,11221 517 1044 1165 1 525_2788_2837|1,12231 525 2788 2837 525_92_533|1,13121 525 92 533 527_336_625|0,13113 527 336 625 527_336_625|1,1222 527 336 625 1 53_1404_1405|1,3131 53 1404 1405 531_1700_1781|0,113211 531 1700 1781 533_756_925|1,21221 533 756 925 5335_3192_6217|0,121213 5335 3192 6217 5341_4740_7141|0,112212 5341 4740 7141 5355_6068_8093|0,121232 5355 6068 8093 539_1140_1261|0,11221 539 1140 1261 5405_5508_7717|0,112122 5405 5508 7717 5439_4960_7361|0,121312 5439 4960 7361 55_1512_1513|1,3133 55 1512 1513 55_48_73|0,112 55 48 73 55_48_73|1,22 55 48 73 1 551_240_601|0,111123 551 240 601 551_240_601|1,1232 551 240 601 1 553_3096_3145|1,32311 553 3096 3145 555_572_797|0,11122 555 572 797 555_572_797|1,13212 555 572 797 1 559_840_1009|0,113331 559 840 1009 559_840_1009|1,3123 559 840 1009 1 5605_2772_6253|0,121323 5605 2772 6253 561_1240_1361|0,12321 561 1240 1361 561_1240_1361|1,22211 561 1240 1361 1 565_6372_6397|1,22331 565 6372 6397 57_1624_1625|1,3311 57 1624 1625 57_176_185|0,1211 57 176 185 57_176_185|1,2111 57 176 185 1 5733_5644_8045|0,131322 5733 5644 8045 575_48_577|1,1332 575 48 577 5757_3476_6725|0,132213 5757 3476 6725 5763_5084_7685|0,123212 5763 5084 7685 5781_6460_8669|0,123232 5781 6460 8669 5785_1848_6073|0,122233 5785 1848 6073 583_1344_1465|1,2223 583 1344 1465 585_2072_2153|1,32211 585 2072 2153 585_928_1097|0,111331 585 928 1097 589_300_661|0,13323 589 300 661 59_1740_1741|1,3313 59 1740 1741 5917_2844_6565|0,131223 5917 2844 6565 6027_9964_11645|0,122231 6027 9964 11645 609_200_641|0,13233 609 200 641 61_1860_1861|1,3331 61 1860 1861 611_1020_1189|0,13231 611 1020 1189 611_1020_1189|1,31213 611 1020 1189 1 615_728_953|0,12332 615 728 953 615_728_953|1,3122 615 728 953 1 615_7552_7577|1,22333 615 7552 7577 621_100_629|1,13321 621 100 629 623_3936_3985|1,12233 623 3936 3985 627_364_725|0,11313 627 364 725 627_364_725|1,21212 627 364 725 1 629_540_829|0,13112 629 540 829 629_540_829|1,22121 629 540 829 1 63_16_65|0,1333 63 16 65 63_16_65|1,32 63 16 65 1 63_1984_1985|1,3333 63 1984 1985 6321_7120_9521|0,132232 6321 7120 9521 637_1116_1285|0,12131 637 1116 1285 6375_6512_9113|0,132122 6375 6512 9113 639_2480_2561|1,11233 639 2480 2561 6405_3652_7373|0,122313 6405 3652 7373 649_1680_1801|0,123311 649 1680 1801 65_2112_2113|1,11111 65 2112 2113 65_72_97|0,132 65 72 97 65_72_97|1,1211 65 72 97 1 651_4300_4349|1,32313 651 4300 4349 663_1216_1385|1,2323 663 1216 1385 663_616_905|0,13312 663 616 905 663_616_905|1,21122 663 616 905 1 665_432_793|0,12113 665 432 793 667_156_685|0,112333 667 156 685 667_156_685|1,2312 667 156 685 1 6695_3192_7417|0,112223 6695 3192 7417 67_2244_2245|1,11113 67 2244 2245 671_1800_1921|0,112311 671 1800 1921 671_1800_1921|1,32123 671 1800 1921 1 675_52_677|1,3112 675 52 677 6765_6052_9077|0,121212 6765 6052 9077 69_2380_2381|1,11131 69 2380 2381 69_260_269|0,13111 69 260 269 69_260_269|1,231 69 260 269 1 693_1924_2045|0,131311 693 1924 2045 693_1924_2045|1,13231 693 1924 2045 1 697_696_985|0,1222 697 696 985 7_24_25|0,111 7 24 25 7_24_25|1,3 7 24 25 1 703_504_865|1,12122 703 504 865 71_2520_2521|1,11133 71 2520 2521 713_216_745|0,111233 713 216 745 715_1428_1597|0,12121 715 1428 1597 7205_7068_10093|0,132322 7205 7068 10093 7239_6440_9689|0,132212 7239 6440 9689 725_108_733|1,31121 725 108 733 73_2664_2665|1,11311 73 2664 2665 731_780_1069|0,111132 731 780 1069 731_780_1069|1,31212 731 780 1069 1 7315_7332_10357|0,113222 7315 7332 10357 735_1088_1313|0,123331 735 1088 1313 735_1088_1313|1,3323 735 1088 1313 1 7383_3344_8105|0,122123 7383 3344 8105 741_1540_1709|0,13221 741 1540 1709 741_1540_1709|1,13221 741 1540 1709 1 741_580_941|1,12321 741 580 941 7421_15540_17221|0,122221 7421 15540 17221 747_3404_3485|1,32213 747 3404 3485 749_5700_5749|1,32331 749 5700 5749 75_2812_2813|1,11313 75 2812 2813 75_308_317|0,12111 75 308 317 75_308_317|1,2113 75 308 317 1 7521_3560_8321|0,123223 7521 3560 8321 759_2320_2441|0,131211 759 2320 2441 759_2320_2441|1,11223 759 2320 2441 1 759_280_809|0,12133 759 280 809 759_280_809|1,12132 759 280 809 1 765_868_1157|0,11232 765 868 1157 765_868_1157|1,12221 765 868 1157 1 767_1656_1825|0,111321 767 1656 1825 77_2964_2965|1,11331 77 2964 2965 77_36_85|0,123 77 36 85 77_36_85|1,321 77 36 85 1 775_168_793|1,21132 775 168 793 7755_8428_11453|0,122132 7755 8428 11453 777_464_905|0,11213 777 464 905 779_660_1021|0,12112 779 660 1021 781_2460_2581|0,112211 781 2460 2581 783_56_785|1,3132 783 56 785 79_3120_3121|1,11333 79 3120 3121 7923_8036_11285|0,123122 7923 8036 11285 793_1776_1945|0,113321 793 1776 1945 795_1292_1517|0,111231 795 1292 1517 795_1292_1517|1,33213 795 1292 1517 1 799_960_1249|0,113332 799 960 1249 799_960_1249|1,3322 799 960 1249 1 8023_3864_8905|0,121223 8023 3864 8905 803_2604_2725|0,123211 803 2604 2725 803_2604_2725|1,22213 803 2604 2725 1 805_348_877|1,23121 805 348 877 81_3280_3281|1,13111 81 3280 3281 8165_4092_9133|0,122323 8165 4092 9133 817_744_1105|0,11312 817 744 1105 819_1900_2069|1,23213 819 1900 2069 83_3444_3445|1,13113 83 3444 3445 8319_8200_11681|0,121322 8319 8200 11681 837_116_845|1,31321 837 116 845 847_7296_7345|1,32333 847 7296 7345 8479_7800_11521|0,122312 8479 7800 11521 85_132_157|0,1331 85 132 157 85_132_157|1,221 85 132 157 1 85_3612_3613|1,13131 85 3612 3613 851_420_949|0,11323 851 420 949 855_832_1193|0,13322 855 832 1193 855_832_1193|1,2322 855 832 1193 1 87_3784_3785|1,13133 87 3784 3785 87_416_425|0,131111 87 416 425 87_416_425|1,233 87 416 425 1 871_2160_2329|1,21223 871 2160 2329 89_3960_3961|1,13311 89 3960 3961 8925_4268_9893|0,132223 8925 4268 9893 893_924_1285|0,111122 893 924 1285 8967_8944_12665|0,131222 8967 8944 12665 897_496_1025|0,133313 897 496 1025 899_60_901|1,3312 899 60 901 9_40_41|0,1111 9 40 41 9_40_41|1,11 9 40 41 1 903_704_1145|1,2222 903 704 1145 909_5060_5141|1,32231 909 5060 5141 91_4140_4141|1,13313 91 4140 4141 91_60_109|0,1113 91 60 109 91_60_109|1,212 91 60 109 1 93_4324_4325|1,13331 93 4324 4325 93_476_485|0,121111 93 476 485 93_476_485|1,2131 93 476 485 1 931_1020_1381|0,13132 931 1020 1381 931_1020_1381|1,33212 931 1020 1381 1 935_1368_1657|1,11123 935 1368 1657 935_3552_3673|1,13233 935 3552 3673 943_576_1105|0,13213 943 576 1105 943_576_1105|1,3222 943 576 1105 1 945_248_977|0,131333 945 248 977 949_2580_2749|0,132311 949 2580 2749 949_2580_2749|1,31231 949 2580 2749 1 95_168_193|0,1131 95 168 193 95_168_193|1,123 95 168 193 1 95_4512_4513|1,13333 95 4512 4513 957_124_965|1,33121 957 124 965 969_1120_1481|0,111332 969 1120 1481 969_1480_1769|0,112331 969 1480 1769 97_4704_4705|1,31111 97 4704 4705 975_2728_2897|0,121311 975 2728 2897 975_448_1073|0,13123 975 448 1073 975_448_1073|1,3232 975 448 1073 1 987_884_1325|0,11212 987 884 1325 989_660_1189|0,131113 989 660 1189 99_20_101|0,13333 99 20 101 99_20_101|1,112 99 20 101 1 99_4900_4901|1,31113 99 4900 4901 9975_6032_11657|0,122213 9975 6032 11657 999_320_1049|0,12233 999 320 1049 999_320_1049|1,2232 999 320 1049 1

added triplets contained in both trees; regards Gangleri

Price's tree[edit]

Price's tree
all generations (0 to 6)
0
1 0 1
1 0 #Plato(0) #Pell(1,0)
n/a
2 0
1
3 4 5
2 1 #Plato(1) #Pythagoras(1) #Pell(2,1)
3 1
4 4
12 2
1,1
5 12 13
3 2 #Plato(2)
5 1
12 9
30 4
1,11
9 40 41
5 4 #Plato(4)
9 1
40 25
90 8
1,111
17 144 145
9 8 #Plato(8)
17 1
144 81
306 16
1,1111
33 544 545
17 16 #Plato(16)
33 1
544 289
1122 32
1,11111
65 2112 2113
33 32 #Plato(32)
65 1
2112 1089
4290 64
1,11112
1155 68 1157
34 1 #Pythagoras(17)
35 33
68 1156
2380 66
1,11113
67 2244 2245
34 33 #Plato(33)
67 1
2244 1156
4556 66
1,1112
323 36 325
18 1 #Pythagoras(9)
19 17
36 324
684 34
1,11121
357 76 365
19 2
21 17
76 361
798 68
1,11122
1007 1224 1585
36 17
53 19
1224 1296
3816 646
1,11123
935 1368 1657
36 19
55 17
1368 1296
3960 646 #p3b
1,1113
35 612 613
18 17 #Plato(17)
35 1
612 324
1260 34
1,11131
69 2380 2381
35 34 #Plato(34
69 1
2380 1225
4830 68
1,11132
1295 72 1297
36 1 #Pythagoras(18)
37 35
72 1296
2664 70
1,11133
71 2520 2521
36 35 #Plato(35)
71 1
2520 1296
5112 70
1,112
99 20 101
10 1 #Pythagoras(5)
11 9
20 100
220 18
1,1121
117 44 125
11 2
13 9
44 121
286 36
1,11211
153 104 185
13 4
17 9
104 169
442 72
1,11212
403 396 565
22 9
31 13
396 484
1364 234
1,11213
315 572 653
22 13
35 9
572 484
1540 234
1,1122
319 360 481
20 9
29 11
360 400
1160 198
1,11221
517 1044 1165
29 18
47 11
1044 841
2726 396
1,11222
1479 880 1721
40 11
51 29
880 1600
4080 638
1,11223
759 2320 2441
40 29
69 11
2320 1600
5520 638
1,1123
279 440 521
20 11
31 9
440 400
1240 198
1,11231
477 1364 1445
31 22
53 9
1364 961
3286 396
1,11232
1519 720 1681
40 9
49 31
720 1600
3920 558
1,11233
639 2480 2561
40 31
71 9
2480 1600
5680 558
1,113
19 180 181
10 9 #Plato(9)
19 1
180 100
380 18
1,1131
37 684 685
19 18 #Plato(18)
37 1
684 361
1406 36
1,11311
73 2664 2665 #c3b1
37 36 #Plato(36)
73 1
2664 1369
5402 72
1,11312
1443 76 1445
38 1 #Pythagoras(19)
39 37
76 1444
2964 74
1,11313
75 2812 2813
38 37 #Plato(37)
75 1
2812 1444
5700 74 #p2b
1,1132
399 40 401
20 1 #Pythagoras(10)
21 19
40 400
840 38
1,11321
437 84 445
21 2
23 19
84 441
966 76
1,11322
1239 1520 1961
40 19
59 21
1520 1600
4720 798
1,11323
1159 1680 2041
40 21
61 19
1680 1600
4880 798
1,1133
39 760 761
20 19 #Plato(19)
39 1
760 400
1560 38
1,11331
77 2964 2965
39 38 #Plato(38)
77 1
2964 1521
6006 76
1,11332
1599 80 1601
40 11 #Pythagoras(20)
41 39
80 1600
3280 78
1,11333
79 3120 3121
40 39 #Plato(39)
79 1
3120 1600
6320 78
1,12
35 12 37
6 1 #Pythagoras(3)
7 5
12 36
84 10
1,121
45 28 53
7 2
9 5
28 49
126 20
1,1211
65 72 97
9 4
13 5
72 81
234 40
1,12111
105 208 233
13 8
21 5
208 169
546 80
1,12112
299 180 349
18 5
23 13
180 324
828 130
1,12113
155 468 493
18 13
31 5
468 324
1116 130
1,1212
171 140 221
14 5
19 9
140 196
532 90
1,12121
261 380 461
19 10
29 9
380 361
1102 180
1,12122
703 504 865
28 9
37 19
504 784
2072 342
1,12123
423 1064 1145
28 19
47 9
1064 784
2632 342
1,1213
115 252 277
14 9
23 5
252 196
644 90
1,12131
205 828 853
23 18
41 5
828 529
1886 180
1,12132
759 280 809
28 5
33 23
280 784
1848 230
1,12133
255 1288 1313
28 23
51 5
1288 784
2856 230
1,122
119 120 169
12 5 #Pell(4,3)
17 7
120 144
408 70
1,1221
189 340 389
17 10
27 7
340 289
918 140
1,12211
329 1080 1129
27 20
47 7
1080 729
2538 280
1,12212
1107 476 1205
34 7
41 27
476 1156
2788 378
1,12213
427 1836 1885
34 27
61 7
1836 1156
4148 378
1,1222
527 336 625
24 7
31 17
336 576
1488 238
1,12221
765 868 1157
31 14
45 17
868 961
2790 476
1,12222
2015 1632 2593
48 17
65 31
1632 2304
6240 1054
1,12223
1343 2976 3265
48 31
79 17
2976 2304
7584 1054
1,1223
287 816 865
24 17
41 7
816 576
1968 238
1,12231
525 2788 2837
41 34
75 7
2788 1681
6150 476
1,12232
2255 672 2353
48 7
55 41
672 2304
5280 574
1,12233
623 3936 3985
48 41
89 7
3936 2304
8544 574
1,123
95 168 193
12 7
19 5
168 144
456 70
1,1231
165 532 557
19 14
33 5
532 361
1254 140
1,12311
305 1848 1873
33 28
61 5
1848 1089
4026 280
1,12312
1419 380 1469
38 5
43 33
380 1444
3268 330
1,12313
355 2508 2533
38 33
71 5
2508 1444
5396 330
1,1232
551 240 601
24 5
29 19
240 576
1392 190
1,12321
741 580 941
29 10
39 19
580 841
2262 380
1,12322
1943 1824 2665 #c3b1
48 19
69 29
1824 2304
6432 1102
1,12323
1463 2784 3145
48 29
77 19
2784 2304
7392 1102
1,1233
215 912 937
24 19
43 5
912 576
2064 190
1,12331
405 3268 3293
43 38
81 5
3268 1849
6966 380
1,12332
2279 480 2329
48 5
53 43
480 2304
5088 430
1,12333
455 4128 4153
48 43
91 5
4128 2304
8736 430
1,13
11 60 61
6 5 #Plato(5)
11 1
60 36
132 10
1,131
21 220 221
11 10 #Plato(10)
21 1
220 121
462 20
1,1311
41 840 841
21 20 #Plato(20)
41 1
840 441
1722 40
1,13111
81 3280 3281
41 40 #Plato(40)
81 1
3280 1681
6642 80
1,13112
1763 84 1765
42 1 #Pythagoras(21)
43 41
84 1764
3612 82
1,13113
83 3444 3445
42 41 #Plato(41)
83 1
3444 1764
6972 82
1,1312
483 44 485
22 1 #Pythagoras(11)
23 21
44 484
1012 42
1,13121
525 92 533
23 2
25 21
92 529
1150 84
1,13122
1495 1848 2377
44 21
65 23
1848 1936
5720 966 #p5b
1,13123
1407 2024 2465
44 23
67 21
2024 1936
5896 966
1,1313
43 924 925
22 21 #Plato(21)
43 1
924 484
1892 42
1,13131
85 3612 3613
43 42 #Plato(42)
85 1
3612 1849
7310 84
1,13132
1935 88 1937
44 1 #Pythagoras(22)
45 43
b n
3960 86 #p3b
1,13133
87 3784 3785
44 43 #Plato(43)
87 1
88 1936
7656 86
1,132
143 24 145
12 1 #Pythagoras(6)
13 11
24 144
312 22
1,1321
165 52 173
13 2
15 11
b n
390 44
1,13211
209 120 241
15 4
19 11
b n
570 88
1,13212
555 572 797
26 11
37 15
b n
1924 330
1,13213
451 780 901
26 15
41 11
b n
2132 330
1,1322
455 528 697
24 11
53 13
b n
1680 286
1,13221
741 1540 1709
35 22
57 13
b n
3990 572
1,13222
2135 1248 2473
48 13
61 53
b n
5856 910
1,13223
1079 3360 3529
48 35
83 13
b n
7968 910
1,1323
407 624 745
24 13
37 11
b n
1776 286
1,13231
693 1924 2045
37 26
63 11
b n
4662 572
1,13232
2183 1056 2425
48 11
59 37
b n
5664 814
1,13233
935 3552 3673
48 37
85 11
b n
8160 814
1,133
23 264 265
12 11 #Plato(11)
23 1
b n
552 22
1,1331
45 1012 1013
23 22 #Plato(22)
45 1
b n
2070 44
1,13311
89 3960 3961
45 44 #Plato(44)
89 1
b n
8010 88
1,13312
2115 92 2117
46 1 #Pythagoras(23)
47 45
b n
4324 90
1,13313
91 4140 4141
46 45 #Plato(45)
91 1
b n
8372 90
1,1332
575 48 577
24 1 #Pythagoras(12)
25 23
b n
1200 46
1,13321
621 100 629
25 2
27 23
b n
1350 92
1,13322
1775 2208 2833
48 23
71 25
b n
6816 1150
1,13323
1679 2400 2929
48 25
73 23
b n
7008 1150
1,1333
47 1104 1105
24 23 #Plato(23)
47 1
b n
2256 46
1,13331
93 4324 4325
47 46 #Plato(46)
93 1
b n
8742 92
1,13332
2303 96 2305
48 1 #Pythagoras(24)
49 47
b n
4704 94
1,13333
95 4512 4513
48 47 #Plato(47)
95 1
b n
9120 94
1,2
15 8 17
4 1 #Pythagoras(2)
5 3
b n
40 6
1,21
21 20 29
5 2 #Pell(3,2)
7 3
b n
70 12
1,211
33 56 65
7 4
11 3
b n
154 24
1,2111
57 176 185
11 8
19 3
b n
418 48
1,21111
105 608 617
19 16
35 3
b n
1330 96
1,21112
475 132 493
22 3
25 19
b n
1100 114
1,21113
123 836 845
22 19
41 3
b n
1804 114
1,2112
187 84 205
14 3
17 11
b n
476 66
1,21121
253 204 325
17 6
23 11
b n
782 132
1,21122
663 616 905
28 11
39 17
b n
2184 374
1,21123
495 952 1073
28 17
45 11
b n
2520 374
1,2113
75 308 317
14 11
25 3
b n
700 66
1,21131
141 1100 1109
25 22
47 3
b n
2350 132
1,21132
775 168 793
28 3
31 25
b n
1736 150
1,21133
159 1400 1409
28 25
53 3
b n
2968 150
1,212
91 60 109
10 3
13 7
b n
260 42
1,2121
133 156 205
13 6
19 7
b n
494 84
1,21211
217 456 505
19 12
31 7
b n
1178 168
1,21212
627 364 725
26 7
33 19
b n
1716 266 #p1b
1,21213
315 988 1037
26 19
45 7
b n
2340 266
1,2122
351 280 449
20 7
27 13
b n
1080 182
1,21221
533 756 925
27 14
41 13
b n
2214 364
1,21222
1431 1040 1769
40 13
53 27
b n
4240 702
1,21223
871 2160 2329
40 27
67 13
b n
5360 702
1,2123
231 520 569
20 13
33 7
b n
1320 182
1,21231
413 1716 1765
33 26
59 7
b n
3894 364
1,21232
1551 560 1649
40 7
47 33
b n
3760 462
1,21233
511 2640 2689
40 33
73 7
b n
5840 462
1,213
51 140 149
10 7
17 3
b n
340 42
1,2131
93 476 485
17 14
31 3
b n
1054 84
1,21311
177 1736 1745
31 28
59 3
b n
3658 168
1,21312
1147 204 1165
34 3
37 31
b n
2516 186
1,21313
195 2108 2117
34 31
65 3
b n
4420 186
1,2132
391 120 409
20 3
23 17
b n
920 102
1,21321
493 276 565
23 6
29 17
b n
1334 204
1,21322
1311 1360 1889
40 17
57 23
b n
4560 782
1,21323
1071 1840 2129
40 23
63 17
b n
5040 782
1,2133
111 680 689
20 17
37 3
b n
1480 102
1,21331
213 2516 2525
37 34
71 3
b n
5254 204
1,21332
1591 240 1609
40 3
43 37
b n
3440 222
1,21333
231 2960 2969
40 37
77 3
b n
6160 222
1,22
55 48 73
8 3
11 5
b n
176 30
1,221
85 132 157
11 6
17 5
b n
374 60
1,2211
145 408 433
17 12
29 5
b n
986 120
1,22111
265 1392 1417
29 24
53 5
b n
3074 240
1,22112
1131 340 1181
34 5
39 29
b n
2652 290 #p2b
1,22113
315 1972 1997
34 29
63 5
b n
4284 290
1,2212
459 220 509
22 5
27 17
b n
1188 170
1,22121
629 540 829
27 10
37 17
b n
1998 340
1,22122
1647 1496 2225
44 17
61 27
b n
5368 918
1,22123
1207 2376 2665 #c3b1
44 27
71 17
b n
6248 918
1,2213
195 748 773
22 17
39 5
b n
1716 170 #p1b
1,22131
365 2652 2677
39 34
73 5
b n
5694 340
1,22132
1911 440 1961
44 5
49 39
b n
4312 390
1,22133
415 3432 3457
44 39
83 5
b n
7304 390
1,222
231 160 281
16 5
21 11
b n
672 110
1,2221
341 420 541
21 10
31 11
b n
1302 220
1,22211
561 1240 1361
31 20
51 11
b n
3162 440
1,22212
1643 924 1885
42 11
53 31
b n
4452 682
1,22213
803 2604 2725
42 31
73 11
b n
6132 682
1,2222
903 704 1145
32 11
43 21
b n
2752 462
1,22221
1365 1892 2333
43 22
65 21
b n
5590 924
1,22222
3655 2688 4537
64 21
85 43
b n
10880 1806
1,22223
2247 5504 5945
64 43
107 21
b n
13696 1806
1,2223
583 1344 1465
32 21
53 11
b n
3392 462
1,22231
1045 4452 4573
53 42
95 11
b n
10070 924
1,22232
3975 1408 4217
64 11
75 53
b n
9600 1166
1,22233
1287 6784 6905
64 53
117 11
b n
14976 1166
1,223
135 352 377
16 11
27 5
b n
864 110
1,2231
245 1188 1213
27 22
49 5
b n
2646 220
1,22311
465 4312 4337
49 44
93 5
b n
9114 440
1,22312
2891 540 2941
54 5
59 49
b n
6372 490
1,22313
515 5292 5317
54 49
103 5
b n
11124 490
1,2232
999 320 1049
32 5
37 27
b n
2368 270
1,22321
1269 740 1469
37 10
47 27
b n
3478 540
1,22322
3367 3456 4825
64 27
91 37
b n
11648 1998
1,22323
2727 4736 5465
64 37
101 27
b n
12928 1998
1,2233
295 1728 1753
32 27
59 5
b n
3776 270
1,22331
565 6372 6397
59 54
113 5
b n
13334 540
1,22332
4071 640 4121
64 5
69 59
b n
8832 590
1,22333
615 7552 7577
64 59
123 5
b n
15744 590
1,23
39 80 89
8 5
13 3
b n
208 30
1,231
69 260 269
13 10
23 3
b n
598 60
1,2311
129 920 929
23 20
43 3
b n
1978 120
1,23111
249 3440 3449
43 40
83 3
b n
7138 240
1,23112
2107 276 2125
46 3
49 43
b n
4508 258
1,23113
267 3956 3965
46 43
89 3
b n
8188 258
1,2312
667 156 685
26 3
29 23
b n
1508 138
1,23121
805 348 877
29 6
35 23
b n
2030 276
1,23122
2175 2392 3233
52 23
75 29
b n
7800 1334
1,23123
1863 3016 3545
52 29
81 23
b n
8424 1334
1,2313
147 1196 1205
26 23
49 3
b n
2548 138
1,23131
285 4508 4517
49 46
95 3
b n
9310 276
1,23132
2695 312 2713
52 3
55 49
b n
5720 294 #p5b
1,23133
303 5096 5105
52 49
101 3
b n
10504 294
1,232
247 96 265
16 3
19 13
b n
608 78
1,2321
325 228 397
19 6
25 13
b n
950 156
1,23211
481 600 769
25 12
37 13
b n
1850 312
1,23212
1275 988 1613
38 13
51 v
b n
3876 650
1,23213
819 1900 2069
38 25
63 13
b n
4788 650
1,2322
855 832 1193
32 13
45 19
b n
2880 494
1,23221
1349 2340 2701
45 26
71 19
b n
6390 988
1,23222
3735 2432 4457
64 19
83 45
b n
10624 1710
1,23223
2071 5760 6121
64 45
109 19
b n
13952 1710
1,2323
663 1216 1385
32 19
51 13
b n
3264 494
1,23231
1157 3876 4045
51 38
89 13
b n
9078 988
1,23232
3927 1664 4265
64 13
77 51
b n
9856 1326
1,23233
1495 6528 6697
64 51
115 13
b n
14720 1326
1,233
87 416 425
16 13
29 3
b n
928 78
1,2331
165 1508 1517
29 26
55 3
b n
3190 156
1,23311
321 5720 5729
55 52
107 3
b n
11770 312
1,23312
3355 348 3373
58 3
61 55
b n
7076 330
1,23313
339 6380 6389
58 55
113 3
b n
13108 330
1,2332
1015 192 1033
32 3
35 29
b n
2240 174
1,23321
1189 420 1261
35 6
41 29
b n
2870 348
1,23322
3255 3712 4937
64 29
93 35
b n
11904 2030
1,23323
2871 4480 5321
64 35
99 29
b n
12672 2030
1,2333
183 1856 1865
32 29
61 3
b n
3904 174
1,23331
357 7076 7085
61 58
119 3
b n
14518 348
1,23332
4087 384 4105
64 3
67 61
b n
8576 366
1,23333
375 7808 7817
64 61
125 3
b n
16000 366
1,3
7 24 25
4 3 #Plato(3)
7 1
b n
56 6
1,31
13 84 85
7 6 #Plato(6)
13 1
b n
182 12
1,311
25 312 313
13 12 #Plato(12)
25 1
b n
650 24
1,3111
49 1200 1201
25 24 #Plato(24)
49 1
b n
2450 48
1,31111
97 4704 4705
49 48 #Plato(48)
97 1
b n
9506 96
1,31112
2499 100 2501
50 1 #Pythagoras(25)
51 49
b n
5100 98
1,31113
99 4900 4901
50 49;nbsp;#Plato(49)
99 1
b n
9900 98
1,3112
675 52 677
26 1 #Pythagoras(13)
27 25
b n
1404 50
1,31121
725 108 733
27 2
29 25
b n
1566 100
1,31122
2079 2600 3329
52 25
77 27
b n
8008 1350
1,31123
1975 2808 3433
52 27
79 25
b n
8216 1350
1,3113
51 1300 1301
26 25 #Plato(25)
51 1
b n
2652 50 #p2b
1,31131
101 5100 5101
51 50 #Plato(50)
101 1
b n
10302 100
1,31132
2703 104 2705
52 1 #Pythagoras(26)
53 51
b n
5512 102
1,31133
103 5304 5305
52 51 #Plato(51)
103 1
b n
10712 102
1,312
195 28 197
14 1 #Pythagoras(7)
15 13
b n
420 26
1,3121
221 60 229
15 2
17 13
b n
510 52
1,31211
273 136 305
17 4
21 13
b n
714 104
1,31212
731 780 1069
30 13
43 17
b n
2580 442
1,31213
611 1020 1189
30 17
47 13
b n
2820 442
1,3122
615 728 953
28 13
41 15
b n
2296 390
1,31221
1005 2132 2357
41 26
67 15
b n
5494 780
1,31222
2911 1680 3361
56 15
71 41
b n
7952 1230
1,31223
1455 4592 4817
56 41
97 15
b n
10864 1230
1,3123
559 840 1009
28 15
43 13
b n
2408 390
1,31231
949 2580 2749
43 30
73 13
b n
6278 780
1,31232
2967 1456 3305
56 13
69 43
b n
7728 1118
1,31233
1287 4816 4985
56 43
99 13
b n
11088 1118
1,313
27 364 365
14 13 #Plato(13)
27 1
b n
756 26
1,3131
53 1404 1405
27 26 #Plato(26)
53 1
b n
2862 52
1,31311
105 5512 5513
53 52 #Plato(52)
105 1
b n
11130 104
1,31312
2915 108 2917
54 1 #Pythagoras(27)
55 53
b n
5940 106
1,31313
107 5724 5725
54 53 #Plato(53)
107 1
b n
11556 106
1,3132
783 56 785
28 1 #Pythagoras(14)
29 27
b n
1624 54
1,31321
837 116 845
29 2
31 29
b n
1798 108
1,31322
2407 3024 3865
56 27
83 27
b n
9296 1566
1,31323
2295 3248 3977
56 29
85 29
b n
9520 1566
1,3133
55 1512 1513
28 27 #Plato(27)
55 1
b n
3080 54
1,31331
109 5940 5941
55 54 #Plato(54)
109 1
b n
11990 108
1,31332
3135 112 3137
56 1 #Pythagoras(28)
57 55
b n
6384 110
1,31333
111 6160 6161
56 55 #Plato(55)
111 1
b n
12432 110
1,32
63 16 65
8 1 #Pythagoras(4)
9 7
b n
144 14
1,321
77 36 85
9 2
11 7
b n
198 28
1,3211
105 88 137
11 4
15 7
b n
330 56
1,32111
161 240 289
15 8
23 7
b n
690 112
1,32112
435 308 533
22 7
29 15
b n
1276 210
1,32113
259 660 709
22 15
37 7
b n
1628 210
1,3212
275 252 373
18 7
25 11
b n
900 154
1,32121
429 700 821
25 14
39 11
b n
1950 308
1,32122
1175 792 1417
36 11
47 25
b n
3384 550
1,32123
671 1800 1921
36 25
61 11
b n
4392 550
1,3213
203 396 445
18 11
29 7
b n
1044 154
1,32131
357 1276 1325
29 22
51 7
b n
2958 308
1,32132
1247 504 1345
36 7
43 29
b n
3096 406
1,32133
455 2088 2137
36 29
65 7
b n
4680 406
1,322
207 224 305
16 7
23 9
b n
736 126
1,3221
333 644 725
23 14
37 9
b n
1702 252
1,32211
585 2072 2153
37 28
65 9
b n
4810 504
1,32212
2035 828 2197
46 9
55 37
b n
5060 666
1,32213
747 3404 3485
46 37
83 9
b n
7636 666
1,3222
943 576 1105
32 9
41 23
b n
2624 414
1,32221
1357 1476 2005
41 18
59 23
b n
4838 828
1,32222
3567 2944 4625
64 23
97 41
b n
11136 1886
1,32223
2415 5248 5777
64 41
105 23
b n
13440 1886
1,3223
495 1472 1553
32 23
55 9
b n
3520 414
1,32231
909 5060 5141
55 46
101 9
b n
11110 828
1,32232
4015 1152 4177
64 9
73 55
b n
9344 990
1,32233
1071 7040 7121
64 55
119 9
b n
15232 990
1,323
175 288 337
16 9
25 7
b n
800 126
1,3231
301 900 949
25 18
43 7
b n
2150 252
1,32311
553 3096 3145
43 36
79 7
b n
6794 504
1,32312
2451 700 2549
50 7
57 43
b n
5700 602 #p4b
1,32313
651 4300 4349
50 43
93 7
b n
9300 602
1,3232
975 448 1073
32 7
39 25
b n
2496 350
1,32321
1325 1092 1717
39 14
53 25
b n
4134 700
1,32322
3471 3200 4721
64 25
89 39
b n
11392 1950
1,32323
2575 4992 5617
64 39
103 25
b n
13184 1950
1,3233
399 1600 1649
32 25
57 7
b n
3648 350
1,32331
749 5700 5749
57 50
107 7
b n
12198 700
1,32332
4047 896 4145
64 7
71 57
b n
9088 798
1,32333
847 7296 7345
64 57
121 7
b n
15488 798
1,33
15 112 113
8 7 #Plato(7)
15 1
b n
240 14
1,331
29 420 421
15 14 #Plato(14)
29 1
b n
870 28
1,3311
57 1624 1625
29 28 #Plato(28)
57 1
b n
3306 56
1,33111
113 6384 6385
57 56 #Plato(56)
113 1
b n
12882 112
1,33112
3363 116 3365
58 1 #Pythagoras(29)
59 57
b n
6844 114
1,33113
115 6612 6613
58 57 #Plato(57)
115 1
b n
13340 114
1,3312
899 60 901
30 1 #Pythagoras(15)
31 29
b n
1860 58
1,33121
957 124 965
31 2
33 29
b n
2046 116
1,33122
2759 3480 4441
60 29
89 31
b n
10680 1798
1,33123
2639 3720 4561
60 31
91 29
b n
10920 1798
1,3313
59 1740 1741
30 29 #Plato(29)
59 1
b n
3540 58
1,33131
117 6844 6845
59 58 #Plato(58)
117 1
b n
13806 116
1,33132
3599 120 3601
60 1 #Pythagoras(30)
61 59
b n
7320 118
1,33133
119 7080 7081
60 59 #Plato(59)
119 1
b n
14280 118
1,332
255 32 257
16 1 #Pythagoras(8)
17 15
b n
544 30
1,3321
285 68 293
17 2
19 15
b n
646 60
1,33211
345 152 377
19 4
23 15
b n
874 120
1,33212
931 1020 1381
34 15
49 19
b n
3332 570
1,33213
795 1292 1517
34 19
53 15
b n
3604 570
1,3322
799 960 1249
32 15
47 17
b n
3008 510
1,33221
1309 2820 3109
47 30
77 17
b n
7238 1020
1,33222
3807 2176 4385
64 17
81 47
b n
10368 1598
1,33223
1887 6016 6305
64 47
b n
111 17
14208 1598
1,3323
735 1088 1313
32 17
49 15
b n
3136 510
1,33231
1245 3332 3557
49 34
83 15
b n
8134 1020
1,33232
3871 1920 4321
64 15
79 49
b n
10112 1470
1,33233
1695 6272 6497
64 49
113 15
b n
14464 1470
1,333
31 480 481
16 15 #Plato(17)
31 1
b n
992 30
1,3331
61 1860 1861
31 30 #Plato(30)
61 1
b n
3782 60
1,33311
121 7320 7321
61 60 #Plato(60)
121 1
b n
14762 120
1,33312
3843 124 3845
62 1 #Pythagoras(31)
63 61
b n
7812 122
1,33313
123 7564 7565
62 61 #Plato(61)
123 1
b n
15252 122
1,3332
1023 64 1025
32 1 #Pythagoras(16)
33 31
b n
2112 62
1,33321
1085 132 1093
33 2
35 31
b n
2310 124
1,33322
3135 3968 5057
64 31
95 33
b n
12160 2046
1,33323
3007 4224 5185
64 33
97 31
b n
12416 2046
1,3333
63 1984 1985
32 31 #Plato(31)
63 1
b n
4032 62
1,33331
125 7812 7813
63 62 #Plato(62)
125 1
b n
15750 124
1,33332
4095 128 4097
64 1 #Pythagoras(32)
65 63
b n
8320 126
1,33333
127 8064 8065
64 63 #Plato(63)
127 1
b n
16256 126
top of table


details to Price's tree[edit]

to be continued 2A02:810D:41C0:134:6C5D:2ACF:DA6:D533 (talk) 18:07, 12 July 2019 (UTC)


sorted (b, n) pairs for the Price's tree

0 1 4 4 12 9 40 25 144 81 544 289 2112 1089 68 1156 2244 1156 36 324 76 361 1224 1296 1368 1296 612 324 2380 1225 72 1296 2520 1296 20 100 44 121 104 169 396 484 572 484 360 400 1044 841 880 1600 2320 1600 440 400 1364 961 720 1600 2480 1600 180 100 684 361 2664 1369 76 1444 2812 1444 40 400 84 441 1520 1600 1680 1600 760 400 2964 1521 80 1600 3120 1600 12 36 28 49 72 81 208 169 180 324 468 324 140 196 380 361 504 784 1064 784 252 196 828 529 280 784 1288 784 120 144 340 289 1080 729 476 1156 1836 1156 336 576 868 961 1632 2304 2976 2304 816 576 2788 1681 672 2304 3936 2304 168 144 532 361 1848 1089 380 1444 2508 1444 240 576 580 841 1824 2304 2784 2304 912 576 3268 1849 480 2304 4128 2304 60 36 220 121 840 441 3280 1681 84 1764 3444 1764 44 484 92 529 1848 1936 2024 1936 924 484 3612 1849 88 1936 3784 1936 24 144 52 169 120 225 572 676 780 676 528 576 1540 1225 1248 2304 3360 2304 624 576 1924 1369 1056 2304 3552 2304 264 144 1012 529 3960 2025 92 2116 4140 2116 48 576 100 625 2208 2304 2400 2304 1104 576 4324 2209 96 2304 4512 2304 8 16 20 25 56 49 176 121 608 361 132 484 836 484 84 196 204 289 616 784 952 784 308 196 1100 625 168 784 1400 784 60 100 156 169 456 361 364 676 988 676 280 400 756 729 1040 1600 2160 1600 520 400 1716 1089 560 1600 2640 1600 140 100 476 289 1736 961 204 1156 2108 1156 120 400 276 529 1360 1600 1840 1600 680 400 2516 1369 240 1600 2960 1600 48 64 132 121 408 289 1392 841 340 1156 1972 1156 220 484 540 729 1496 1936 2376 1936 748 484 2652 1521 440 1936 3432 1936 160 256 420 441 1240 961 924 1764 2604 1764 704 1024 1892 1849 2688 4096 5504 4096 1344 1024 4452 2809 1408 4096 6784 4096 352 256 1188 729 4312 2401 540 2916 5292 2916 320 1024 740 1369 3456 4096 4736 4096 1728 1024 6372 3481 640 4096 7552 4096 80 64 260 169 920 529 3440 1849 276 2116 3956 2116 156 676 348 841 2392 2704 3016 2704 1196 676 4508 2401 312 2704 5096 2704 96 256 228 361 600 625 988 1444 1900 1444 832 1024 2340 2025 2432 4096 5760 4096 1216 1024 3876 2601 1664 4096 6528 4096 416 256 1508 841 5720 3025 348 3364 6380 3364 192 1024 420 1225 3712 4096 4480 4096 1856 1024 7076 3721 384 4096 7808 4096 24 16 84 49 312 169 1200 625 4704 2401 100 2500 4900 2500 52 676 108 729 2600 2704 2808 2704 1300 676 5100 2601 104 2704 5304 2704 28 196 60 225 136 289 780 900 1020 900 728 784 2132 1681 1680 3136 4592 3136 840 784 2580 1849 1456 3136 4816 3136 364 196 1404 729 5512 2809 108 2916 5724 2916 56 784 116 841 3024 3136 3248 3136 1512 784 5940 3025 112 3136 6160 3136 16 64 36 81 88 121 240 225 308 484 660 484 252 324 700 625 792 1296 1800 1296 396 324 1276 841 504 1296 2088 1296 224 256 644 529 2072 1369 828 2116 3404 2116 576 1024 1476 1681 2944 4096 5248 4096 1472 1024 5060 3025 1152 4096 7040 4096 288 256 900 625 3096 1849 700 2500 4300 2500 448 1024 1092 1521 3200 4096 4992 4096 1600 1024 5700 3249 896 4096 7296 4096 112 64 420 225 1624 841 6384 3249 116 3364 6612 3364 60 900 124 961 3480 3600 3720 3600 1740 900 6844 3481 120 3600 7080 3600 32 256 68 289 152 361 1020 1156 1292 1156 960 1024 2820 2209 2176 4096 6016 4096 1088 1024 3332 2401 1920 4096 6272 4096 480 256 1860 961 7320 3721 124 3844 7564 3844 64 1024 132 1089 3968 4096 4224 4096 1984 1024 7812 3969 128 4096 8064 4096

added sorted (b, n) pairs for the Price's tree; regards Gangleri 217.86.249.144 (talk) 14:34, 16 July 2019 (UTC)
updated sorted properly by path no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 12:27, 19 July 2019 (UTC)
getting harassed by a nasty excell sort bug no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 13:16, 19 July 2019 (UTC)
now replaced with values for the Price tree no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 13:28, 19 July 2019 (UTC)
getting harassed by a nasty excell sort bug; using fake number workaround no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 13:50, 19 July 2019 (UTC)

comments[edit]

from user:לערי ריינהארט/sandbox; regards Gangleri 2A02:810D:41C0:134:6C5D:2ACF:DA6:D533 (talk) 18:07, 12 July 2019 (UTC)

like Cronus, Wikipedia devours its children
regards Gangleri 217.86.249.144 (talk) 15:09, 16 July 2019 (UTC)

to do[edit]

consider:

rational triangles a redirect to integer triangles;
rational trigonometry not considered here further
triangles where all trigonometric values of the three angles see also trigonometric functions: sine, cosine, tangent, and their reciprocals cosecant, secant and cotangent are rational numbers; these are Pythagorean triangles redirecting to Pythagorean triple.

the six values for the trigonometric functions: sine, cosine, tangent, and their reciprocals cosecant, secant and cotangent are keys (cryptography) each enabeling to identify a unique primitive Pythagorean triple today a redirect to Pythagorean triple; a better redirect could be tree of primitive Pythagorean triples; the later should be moved to trees of primitive Pythagorean triples; the representation of the rational number values associated with each trigonometric function would be the pair of two integer numbers used to codify the rational number as a fraction
So far 13 keys for each primitive Pythagorean triple are identified. It is a bias issue of each contributor how many and which to be used.
regards Gangleri 95.91.248.133 (talk) 10:29, 17 July 2019 (UTC)

Pythagorean numbers a redirect to Pythagorean prime
Pythagorean pair / Pythagorean pairs referred in file:Pythagorean pairs.svg

regards no bias — קיין ומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contributions 20:47, 17 July 2019 (UTC)