Euler's sum of powers conjecture: Difference between revisions
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'''Euler's conjecture''' is a [[conjecture]] in [[mathematics]] related to [[Fermat's last theorem]] which was proposed by [[Leonhard Euler]] in 1769. It states that for all [[integers]] ''n'' and ''k'' greater than 1, if the sum of ''n'' ''k''th powers of positive integers is itself a ''k''th power, then ''n'' is not smaller than ''k''. |
'''Euler's conjecture''' is a disproved [[conjecture]] in [[mathematics]] related to [[Fermat's last theorem]] which was proposed by [[Leonhard Euler]] in 1769. It states that for all [[integers]] ''n'' and ''k'' greater than 1, if the sum of ''n'' ''k''th powers of positive integers is itself a ''k''th power, then ''n'' is not smaller than ''k''. |
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In symbols, if |
In symbols, if |
Revision as of 23:44, 21 September 2008
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which 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 positive integers is itself a kth power, then n is not smaller than k.
In symbols, if where and are positive integers, then .
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.
In 1986, Noam Elkies found a method to construct counterexamples for the k = 4 case. 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 , if , where are positive integers for all and , then
See also
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.thinkquest.org
- A simple explanation of Euler's Conjecture at Maths Is Good For You!
- Mathematicians Find New Solutions To An Ancient Puzzle