Skip to main content
Mathematics

Return to Answer

Added link to 1966 paper
Source Link
Daniel R
Daniel R
  • 3.3k
  • 3
  • 28
  • 40

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 3$ if $k\ge 3$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. ParkinL. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 3$ if $k\ge 3$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 3$ if $k\ge 3$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Corrected factual inaccuracy.
Source Link
John Gowers
John Gowers
  • 25.8k
  • 6
  • 69
  • 103

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 2$$n\ge 3$ if $k\ge 2$$k\ge 3$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 2$ if $k\ge 2$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 3$ if $k\ge 3$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Made it a lot nicer
Source Link
John Gowers
John Gowers
  • 25.8k
  • 6
  • 69
  • 103

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem that states that forabout the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 2$ if $k\ge 2$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem that states that for the Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

to have any solutions, $n$ must be at least $k$ (FLT is the statement that $n\ge 2$ if $k\ge 2$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$

It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 2$ if $k\ge 2$). For small values of $X_i,Y$, the conjecture appears to be true.

In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:

$$25^5+84^5+110^5+133^5=144^5.$$

In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:

$$95800^4+217519^4+414560^4=422481^4.$$

This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.

Source Link
John Gowers
John Gowers
  • 25.8k
  • 6
  • 69
  • 103
Post Made Community Wiki by John Gowers