Wikipedia:Reference desk/Archives/Mathematics/2023 June 10

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June 10

Other questions about generalized pentagonal numbers

After Wikipedia:Reference desk/Archives/Mathematics/2023 May 12#Sum of the reciprocals of numbers in a set and Wikipedia:Reference desk/Archives/Mathematics/2023 May 20#For which natural numbers n, this sequence contains infinitely many primes?, I also have other questions about the generalized pentagonal numbers:

1. For given natural number n, how many possible residues of a generalized pentagonal numbers mod n?

2. The pentagonal number theorem seems to be not only for the partition function (sequence A000041 in the OEIS), but also for the sum-of-divisors function (sequence A000203 in the OEIS), the only difference is when the last term is $\sigma (0)$, then change it to $n$ instead of 1 or 0, right? If so, please edit the pentagonal number theorem page, see the articles [1] and [2] and [3].

3. Are generalized pentagonal numbers the exponents of $(1-x)(1-x^{2})(1-x^{3})(1-x^{4})\cdots$? If so, which $x^{n}$ have coefficients +1 and which $x^{n}$ have coefficients -1? Are there any coefficient other than 0, +1, and -1?

4. What is the sum of reciprocals of all generalized pentagonal numbers (of course not including 0)? Also, what is the sum of the alternating series of unit fractions with the generalized pentagonal numbers as denominators? And what is the sum of -1/1-1/2+1/5+1/7-1/12-1/15+1/22+1/26-… (the sign of 1/k is the same as the sign of $x^{k}$ in $(1-x)(1-x^{2})(1-x^{3})(1-x^{4})\cdots$)? Also, in Wikipedia:Reference desk/Archives/Mathematics/2023 May 12#Sum of the reciprocals of numbers in a set, I know that the sum of reciprocals of “generalized pentagonal numbers * n + 1” is $-{\frac {8\pi \sin({\frac {\pi }{3}}{\sqrt {\frac {n-24}{n}}})}{{\sqrt {n(n-24)}}(1-2\cos({\frac {\pi }{3}}{\sqrt {\frac {n-24}{n}}}))}}$, but what is the sum of the alternating series of unit fractions with the “generalized pentagonal numbers * n + 1” as denominators? And what is the sum of -1/(n+1)-1/(2*n+1)+1/(5*n+1)+1/(7*n+1)-1/(12*n+1)-1/(15*n+1)+1/(22*n+1)+1/(26*n+1)-… (the sign of 1/(k*n+1) is the same as the sign of $x^{k}$ in $(1-x)(1-x^{2})(1-x^{3})(1-x^{4})\cdots$)?

5. Are there any formula of the sum of the first n generalized pentagonal numbers?

6. All natural numbers are the sum of four squares (Lagrange's four-square theorem), and all natural numbers are the sum of how many (choose the smallest such natural number) generalized pentagonal numbers?

7. I found that “generalized pentagonal numbers * k” or/and “generalized pentagonal numbers * k + 1” is also figurate number for some natural numbers k:

generalized pentagonal numbers * k:

• k = 1: generalized pentagonal numbers
• k = 2: (sequence A152749 in the OEIS) but this sequence seems not to be figurate numbers
• k = 3: Subsequence of the triangular numbers (exactly the triangular numbers divisible by 3)
• k = 4: (sequence A062717 in the OEIS) but this sequence seems not to be figurate numbers
• k = 6: Subsequence of the pronic numbers (exactly the pronic numbers divisible by 6)
• k = 8: Subsequence of the generalized octagonal numbers (exactly the generalized octagonal numbers divisible by 8)
• k = 9: Subsequence of (sequence A045943 in the OEIS) (exactly the numbers in this sequence divisible by 9) but this sequence seems not to be figurate numbers
• k = 12: Subsequence of (sequence A046092 in the OEIS) (exactly the numbers in this sequence divisible by 12) but this sequence seems not to be figurate numbers
• k = 15: Subsequence of (sequence A028895 in the OEIS) (exactly the numbers in this sequence divisible by 15) but this sequence seems not to be figurate numbers
• k = 18: Subsequence of (sequence A028896 in the OEIS) (exactly the numbers in this sequence divisible by 9) but this sequence seems not to be figurate numbers
• k = 24: Subsequence of (sequence A005563 in the OEIS) (exactly the numbers in this sequence divisible by 24) but this sequence seems not to be figurate numbers
• k = 25: Subsequence of (sequence A144065 in the OEIS) (exactly the numbers in this sequence divisible by 25) but this sequence seems not to be figurate numbers
• k = 27: Subsequence of (sequence A000096 in the OEIS) (exactly the numbers in this sequence divisible by 27) but this sequence seems not to be figurate numbers
• k = 32: Subsequence of (sequence A271675 in the OEIS) (exactly the numbers in this sequence divisible by 32) but this sequence seems not to be figurate numbers
• k = 36: Subsequence of (sequence A049598 in the OEIS) (exactly the numbers in this sequence divisible by 36) but this sequence seems not to be figurate numbers

generalized pentagonal numbers * k + 1:

• k = 1: (sequence A319390 in the OEIS) but this sequence seems not to be figurate numbers
• k = 3: Subsequence of (sequence A000124 in the OEIS) (exactly the numbers in this sequence == 1 mod 3) but this sequence seems not to be figurate numbers
• k = 4: Subsequence of (sequence A319384 in the OEIS) (exactly the numbers in this sequence == 1 mod 4) but this sequence seems not to be figurate numbers
• k = 6: Subsequence of (sequence A002061 in the OEIS) (exactly the numbers in this sequence == 1 mod 6) but this sequence seems not to be figurate numbers
• k = 8: Subsequence of (sequence A257083 in the OEIS) (exactly the numbers in this sequence == 1 mod 8) but this sequence seems not to be figurate numbers
• k = 9: Subsequence of the centered triangular numbers (exactly the centered triangular numbers == 1 mod 9)
• k = 12: Subsequence of the centered square numbers (exactly the centered square numbers == 1 mod 12)
• k = 15: Subsequence of the centered pentagonal numbers (exactly the centered pentagonal numbers == 1 mod 15)
• k = 18: Subsequence of the hex numbers (exactly the hex numbers == 1 mod 18)
• k = 21: Subsequence of the centered heptagonal numbers (exactly the centered heptagonal numbers == 1 mod 21)
• k = 24: Subsequence of the square numbers and the centered octagonal numbers (exactly the square numbers == 1 mod 24)
• k = 25: Subsequence of the generalized pentagonal numbers (exactly the generalized pentagonal numbers == 1 mod 25)
• k = 27: Subsequence of the triangular numbers and the centered nonagonal numbers (exactly the triangular numbers == 1 mod 27)
• k = 30: Subsequence of the centered decagonal numbers (exactly the centered decagonal numbers == 1 mod 30)
• k = 32: Subsequence of the generalized octagonal numbers (exactly the generalized octagonal numbers == 1 mod 32)
• k = 36: Subsequence of the star numbers (exactly the star numbers == 1 mod 36)
• k = 49: Subsequence of (sequence A144065 in the OEIS) (exactly the numbers in this sequence == 1 mod 49) but this sequence seems not to be figurate numbers

Do I miss other k such that “generalized pentagonal numbers * k” or/and “generalized pentagonal numbers * k + 1” is also figurate number (of course, “generalized pentagonal numbers * k” cannot be primes, even if k = 1 when the generalized pentagonal number is > 7, and according to Wikipedia:Reference desk/Archives/Mathematics/2023 May 20#For which natural numbers n, this sequence contains infinitely many primes?, “generalized pentagonal numbers * k + 1” contains infinitely many primes assuming Bunyakovsky conjecture is true except k = 24, 25, 27, 32, 49, and for these five values of k this sequence contains no primes)? 210.66.124.167 (talk) 00:57, 10 June 2023 (UTC)

Question 1: see oeis:A290732. In particular, the number of distinct residues for pentagonal numbers (and also generalized pentagonal numbers) mod $n$ is multiplicative, with $a(2^{k})=2^{k},a(3^{k})=3^{k},a(p^{k})=1+\lfloor p^{k+1}/(2p+2)\rfloor$ for prime $p\geq 5$. GalacticShoe (talk) 22:47, 10 June 2023 (UTC)

Question 2: how do you mean? From my limited playing around with the formula, I don't think that $\sigma _{k}(n-1)+\sigma _{k}(n-2)-\sigma _{k}(n-5)-\sigma _{k}(n-7)+\ldots$ seems to yield anything $\sigma _{k}(n)$-related. GalacticShoe (talk) 23:54, 10 June 2023 (UTC)

Please see the articles [4] and [5] and [6]. 210.243.206.107 (talk) 02:41, 12 June 2023 (UTC)

Ah, it does seem to hold specifically for $k=1$ with the base case set to $\sigma (0)=n$. In fact, Divisor function#Other properties and identities already has this written. Feel free to edit the page for the pentagonal number theorem if you want. GalacticShoe (talk) 03:48, 12 June 2023 (UTC)

Question 3: yes, and the coefficients are always $\pm 1$. In particular, for the $k$-th pentagonal number (where $k$ can be negative as per the generalized pentagonal numbers), the coefficient is $(-1)^{k}$; see the formula in pentagonal number theorem. GalacticShoe (talk) 22:50, 10 June 2023 (UTC)

So the sign is +, +, -, -, +, +, -, -, +, +, -, -, …, with period 4? 210.243.206.107 (talk) 02:46, 12 June 2023 (UTC)

Yup. GalacticShoe (talk) 03:48, 12 June 2023 (UTC)

Question 4: $\sum _{k=1}^{\infty }{\frac {1}{(3k^{2}-k)/2}}=3\ln(3)-{\frac {\pi }{\sqrt {3}}}$, and $\sum _{k=-\infty }^{-1}{\frac {1}{(3k^{2}-k)/2}}=6-3\ln(3)-{\frac {\pi }{\sqrt {3}}}$. You may obtain the regular and alternating sum of reciprocals as needed by taking sums and differences of these values. $-1/1-1/2+1/5+1/7-\ldots =6-{\frac {4\pi }{\sqrt {3}}}\approx -1.255197$. GalacticShoe (talk) 00:10, 11 June 2023 (UTC)

In Wikipedia:Reference desk/Archives/Mathematics/2023 May 12#Sum of the reciprocals of numbers in a set, I know that the sum of reciprocals of “generalized pentagonal numbers * n + 1” is $-{\frac {8\pi \sin({\frac {\pi }{3}}{\sqrt {\frac {n-24}{n}}})}{{\sqrt {n(n-24)}}(1-2\cos({\frac {\pi }{3}}{\sqrt {\frac {n-24}{n}}}))}}$, but what is the sum of the alternating series of unit fractions with the “generalized pentagonal numbers * n + 1” as denominators? 210.243.206.107 (talk) 02:40, 12 June 2023 (UTC)

Also you have typos, $3\ln(3)-{\frac {\pi }{\sqrt {3}}}=6-3\ln(3)-{\frac {\pi }{\sqrt {3}}}$? 210.243.206.107 (talk) 02:42, 12 June 2023 (UTC)

Whoops, corrected. GalacticShoe (talk) 03:48, 12 June 2023 (UTC)

Question 6: 3 generalized pentagonal numbers suffice, see Guy, R. K. "Every Number Is Expressible as the Sum of How Many Polygonal Numbers?." Amer. Math. Monthly 101, 169-172, 1994a. In fact, if we were to use only (positive-index) pentagonal numbers, it is conjectured that there are exactly $210$ numbers which require at least $4$ pentagonal numbers (see oeis:A003679), and exactly $6$ which require $5$ (see oeis:A133929.) GalacticShoe (talk) 23:12, 10 June 2023 (UTC)

In general, Fermat stated (and Cauchy proved) that every number is the sum of at most $n$ $n$-gonal; see Fermat polygonal number theorem. $n$ is not always a perfect bound, however.

Gauss, in 1792, proved that every natural number can be written as the sum of three triangular numbers, so for $n=3$ this is the case. See oeis:A061336, and in particular note that the sequence of numbers which require one triangular number (which is the sequence of triangular numbers) is infinite, the sequence of numbers which require two triangular numbers (which includes the sequence of triangular numbers + 1) is infinite, and the sequence of numbers which require three triangular numbers (which includes all numbers congruent to $5$ or $8\,(mod\,9)$) is infinite.

The case of four squares being sufficient and necessary infinitely often is famous, as per the earlier mention of Lagrange's four-square theorem. Three squares also happen infinitely often, as per Fermat's theorem on sums of two squares, as do two squares, thanks to Fermat's theorem on sums of two squares.

The case of pentagonal numbers is interesting, specifically because three suffices if one uses generalized pentagonal numbers, and it appears that all sufficiently large numbers can be decomposed into at most three pentagonal numbers, but there is the aforementioned finite set that requires up to five.

The case of hexagonal number is also interesting, since apparently a number is generalized hexagonal if and only if it is triangular, so precisely three generalized hexagonal numbers suffice (and each number up to three is required infinitely often as per the earlier triangular numbers argument.) However, if one only examines (positive-index) hexagonal numbers, then it again boils down to finite sets of counterexamples, although this time we know for sure that there are finitely many. In particular, Legendre proved in 1830 that every number greater than $1791$ can be written as the sum of at most four hexagonal numbers, while Duke and Schulze-Pillot proved in 1990 that after some sufficiently large number, three suffice. Two numbers, $11$ and $26$, require six hexagonal numbers, while exactly thirteen numbers require at least five (see oeis:A007527.) It is suspected that exactly 638 numbers require at least four (see oeis:A007536.) GalacticShoe (talk) 23:40, 10 June 2023 (UTC)

I know that for positive-index pentagonal numbers, the answer is 5, but what about the generalized pentagonal numbers? 210.243.206.107 (talk) 02:44, 12 June 2023 (UTC)

3, see beginning of above comment. GalacticShoe (talk) 03:50, 12 June 2023 (UTC)

Equation in wikimarkup

Humongous number

Hi all, and happy Saturday to yous. I wonder, can someone translate this equation into mark up fit for an article, please? 0.995 x 107% =4.64 x 10 (that 107% is actually 10 to the power of 32,768, no idea why it came out like that though.) Thanks in advance to all who can help! SN54129 15:19, 10 June 2023 (UTC)

10 to the power 32,768, written out as a decimal number, is a digit 1 followed by a tail of 32,768 digits 0. That is a truly humongous number. Multiplying it by 0.995 replaces the first four digits 1000 by 995, still humongous and nowhere near 4.64 × 10 = 46.4. Where did this equation come from, what is it supposed to represent (an incredibly false statement?), and where should it be used?  --Lambiam 16:45, 10 June 2023 (UTC)

Hi Lambiam, this is the original (reliable!) source for the equation---> thanks for your help! SN54129 18:08, 10 June 2023 (UTC)

If we omit the "×10" we get $0.995^{32768}=4.64\times 10^{-72}$, which is more accurate and more obviously relevant to the problem. —Tamfang (talk) 18:21, 10 June 2023 (UTC)

Instead of LaTeX-style markup you can also use pure wikitext markup:

0.995³²⁷⁶⁸ = 4.64 × 10⁻⁷².

I'm sorry to say, though, that the calculation made in the "reliable" source is bogus. First off, if one could trace all of a person's great¹³grandparents, 15 generations back, their number would be substantially smaller than 2¹⁵; for example, one of the greatgrandfathers of the mother could also be a greatgrandfather of the father, and you would in fact see many branches of the family tree merge as you go back in time, especially to a time when people were much less mobile than they are today and possibly never even travelled outside of their village of birth. Secondly, the 0.5 percent of the population referred to will not be spread uniformly and fully at random over the population, so even if all 15th-generation ancestors were distinct, their probabilities of belonging to the 0.5 percent may not be assumed to be independent.  --Lambiam 20:23, 10 June 2023 (UTC)

Eh, I just realised that when I typed it above, I didn't realise that that 10 at the end is actually to the power of minus 72! D'oh. SN54129

Thanks everyone, interesting stuff! I'll be taking each and everyone's opinion onboard with this. All the best! SN54129 20:30, 10 June 2023 (UTC)

https://www.waterstones.com/blog/family-fortunes-adam-rutherford-on-how-were-all-related-to-royalty has background. It claims to calculate the probability that a Brit born in the 1970s is descended from Edward III (1312–1377). The calculation is based on bad assumptions and I don't think it belongs in Wikipedia. For something more encyclopedic and interesting (at least to non-Brits), Most recent common ancestor#TMRCA of all living humans says: "A study by mathematicians Joseph T. Chang, Douglas Rohde and Steve Olson used a theoretical model to calculate that the MRCA may have lived remarkably recently, possibly as recently as 2,000 years ago." PrimeHunter (talk) 23:52, 10 June 2023 (UTC)

See also Identical ancestors point#Of Homo sapiens for a related question. PrimeHunter (talk) 23:55, 10 June 2023 (UTC)