Euler\'s sum of powers conjecture

Euler's conjecture is a conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is not smaller than k. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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 positive integers, then $n\geq k$ .

The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for k = 5:

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵. The year 1966 in Science and Technology involved some significant events listed below

In 1986, Noam Elkies found a method to construct counterexamples for the k = 4 case. Noam D Elkies (born 1966 in New York City) is a Mathematician. His smallest counterexample was the following:

2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴.

In 1988, Roger Frye subsequently found the smallest possible k = 4 counterexample by a direct computer search using techniques suggested by Elkies:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

In 1966, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that for every $k > 3$ , if $\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k$ , where $a_i\ne b_j$ are positive integers for all $1\leq i\leq n$ and $1\leq j\leq m$ , then $m + n \geq k.$

External links

EulerNet: Computing Minimal Equal Sums Of Like Powers
Euler Quartic Conjecture at MathWorld
Diophantine Equation — 4th Powers at MathWorld
Euler's Conjecture at library. John L Selfridge is an American Mathematician who has contributed to the field of Analytic number theory. Euler's equation of degree four is a mathematical problem proposed by Leonhard Euler in 1772. thinkquest. org
A simple explanation of Euler's Conjecture at Maths Is Good For You!
Mathematicians Find New Solutions To An Ancient Puzzle

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