Euler's sum of powers conjecture

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 many kth powers of positive integers is itself a kth power, then n is greater than or equal to k: _a + _a + ... + _a = b^k ⇒ n ≥ k The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case n = 2: if _a + _a = b^k, then 2 ≥ 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. Euler was aware of the equality 59^4 + 158^4 = 133^4 + 134^4 involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. 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. The general solution of the equation is where a and b are any integers. 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. This was published in a paper comprising just two sentences. 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 1988, Noam Elkies published a method to construct an infinite sequence of counterexamples for the k = 4 case. His smallest counterexample was 26824404 + 153656394 + 187967604 = 206156734. A particular case of Elkies' solutions can be reduced to the identity (85v2 + 484v − 313)4 + (68v2 − 586v + 10)4 + (2u)4 = (357v2 − 204v + 363)4 where u2 = 22030 + 28849v − 56158v2 + 36941v3 − 31790v4.

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Euler's sum of powers conjecture