Euler's sum of powers conjecture

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Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n kth powers of non-zero integers is itself a kth power, then n is greater than or equal to k.

In symbols, if 
\sum_{i=1}^{n} a_i^k = b^k
where n>1 and a_1, a_2, \dots, a_n, b are non-zero integers, then n\geq k.

The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case n = 2: if a_1^k + a_2^k = b^k, then 2 \geq k.

Although the conjecture holds for the case k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4 and k = 5. It is unknown whether the conjecture fails or holds for any value k ≥ 6.

Background[edit]

Euler had an equality for four fourth powers 59^4 + 158^4 = 133^4 + 134^4, this however is not a counterexample. He also provided a complete solution to the four cubes problem as in Plato's number 3^3+4^3+5^3=6^3 or the taxicab number 1729.[1][2] The general solution for:

x_1^3+x_2^3=x_3^3+x_4^3

is

x_1 = 1-(a-3b)(a^2+3b^2), x_2 = (a+3b)(a^2+3b^2)-1
x_3 = (a+3b)-(a^2+3b^2)^2, x_4 = (a^2+3b^2)^2-(a-3b)

where a and b are any integers.

Counterexamples[edit]

Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for k = 5.[3] A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:

275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966),
(−220)5 + 50275 + 62375 + 140685 = 141325 (Scher & Seidl, 1996), and
555 + 31835 + 289695 + 852825 = 853595 (Frye, 2004).

In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the k = 4 case.[4] His smallest counterexample was

26824404 + 153656394 + 187967604 = 206156734.

A particular case of Elkies' solutions can be reduced to the identity[5][6]

(85v2 + 484v − 313)4 + (68v2 − 586v + 10)4 + (2u)4 = (357v2 − 204v + 363)4

where

u2 = 22030 + 28849v − 56158v2 + 36941v3 − 31790v4.

This is an elliptic curve with a rational point at v1 = −31/467. From this initial rational point, one can compute an infinite collection of others. Substituting v1 into the identity and removing common factors gives the numerical example cited above.

In 1988, Roger Frye found the smallest possible counterexample

958004 + 2175194 + 4145604 = 4224814

for k = 4 by a direct computer search using techniques suggested by Elkies.[citation needed] This solution is the only one with values of the variables below 1,000,000.[citation needed]

Generalizations[edit]

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[7] that if k > 3 and \sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k, where ai ≠ bj are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m+n ≥ k. In the special case m = 1, the conjecture states that if

\sum_{i=1}^{n} a_i^k = b^k

(under the conditions given above) then n ≥ k − 1.

The special case may be described as the problem of giving a partition of a perfect power into few like powers. For k = 4, 5, 7, 8 and n = k or k − 1, there are many known solutions. Some of these are listed below. There are no solutions for k = 6 where b ≤ 272580.[citation needed]

k = 4[edit]

958004 + 2175194 + 4145604 = 4224814 (R. Frye, 1988)[citation needed]
304 + 1204 + 2724 + 3154 = 3534 (R. Norrie, 1911)[citation needed]

k = 5[edit]

275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966)
75 + 435 + 575 + 805 + 1005 = 1075 (Sastry, 1934, third smallest[citation needed])

k = 7[edit]

1277 + 2587 + 2667 + 4137 + 4307 + 4397 + 5257 = 5687 (M. Dodrill, 1999)[citation needed]

k = 8[edit]

908 + 2238 + 4788 + 5248 + 7488 + 10888 + 11908 + 13248 = 14098 (S. Chase, 2000)[citation needed]

See also[edit]

References[edit]

  1. ^ William Dunham, ed. (2007). The Genius of Euler: Reflections on His Life and Work. The MAA. p. 220. ISBN 978-0-88385-558-4. 
  2. ^ Titus Piezas III (2005). "Euler's Extended Conjecture". 
  3. ^ L. J. Lander, T. R. Parkin (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72: 1079. doi:10.1090/S0002-9904-1966-11654-3. 
  4. ^ Noam Elkies (1988). "On A4 + B4 + C4 = D4". Mathematics of Computation 51 (184): 825–835. doi:10.2307/2008781. JSTOR 2008781. MR 0930224. 
  5. ^ "Elkies' a^4+b^4+c^4 = d^4". 
  6. ^ "Sums of Three Fourth Powers". 
  7. ^ L. J. Lander, T. R. Parkin, J. L. Selfridge (1967). "A Survey of Equal Sums of Like Powers". Mathematics of Computation 21 (99): 446–459. doi:10.1090/S0025-5718-1967-0222008-0. JSTOR 2003249. 

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