Postscript.

$$(A, B, C;D) = (2682440,15365639,18796760; 20615673)$$ to Euler's conjecture still seems beyond the range of reasonable exhaustive computer search, there remained the possibility that smaller solutions may be found by such a search. Shortly after hearing of the first solution, Roger Frye of Thinking Machines Corporation asked whether it was minimal; I did not know, but suggested how one might exhaustively search for smaller solutions: eliminating common factors and permuting \$A\$, \$B\$, \$C\$ if necessary, we may take \$D\$ odd and not divisible by 5, and \$C < D\$ such that \$D^4 - C^4\$ is divisible by 625 and satisfies several other congruence and divisibility properties, and for each such \$D\$ and \$C\$ look for a representation of \$D^4 — C^4\$ as \$A^4 + B^4\$ with \$A\$, \$B\$ divisible by 5. Frye translated this into a computer program and ran it on various Connection Machines for about 100 hours to find the minimal counterexample to Euler's conjecture: $$95800^4 + 217519^4 + 414560^4 = 422481^4.$$ He continued the search and found that this solution is unique in the range \$D < 10^6\$. This solution appears on the parameterization (6) with \$(m, n) = (20,-9)\$. We include Frye's result with his permission.

Postscript.

$$(A, B, C;D) = (2682440,15365639,18796760; 20615673)$$ to Euler's conjecture still seems beyond the range of reasonable exhaustive computer search, there remained the possibility that smaller solutions may be found by such a search. Shortly after hearing of the first solution, Roger Frye of Thinking Machines Corporation asked whether it was minimal; I did not know, but suggested how one might exhaustively search for smaller solutions: eliminating common factors and permuting \$A\$, \$B\$, \$C\$ if necessary, we may take \$D\$ odd and not divisible by 5, and \$C < D\$ such that \$D^4 - C^4\$ is divisible by 625 and satisfies several other congruence and divisibility properties, and for each such \$D\$ and \$C\$ look for a representation of \$D^4 — C^4\$ as \$A^4 + B^4\$ with \$A\$, \$B\$ divisible by 5. Frye translated this into a computer program and ran it on various Connection Machines for about 100 hours to find the minimal counterexample to Euler's conjecture: $$95800^4 + 217519^4 + 414560^4 = 422481^4.$$ He continued the search and found that this solution is unique in the range \$D < 10^6\$. This solution appears on the parameterization (6) with \$(m, n) = (20,-9)\$. We include Frye's result with his permission.

Postscript.

$$(A, B, C;D) = (2682440,15365639,18796760; 20615673)$$ to Euler's conjecture still seems beyond the range of reasonable exhaustive computer search, there remained the possibility that smaller solutions may be found by such a search. Shortly after hearing of the first solution, Roger Frye of Thinking Machines Corporation asked whether it was minimal; I did not know, but suggested how one might exhaustively search for smaller solutions: eliminating common factors and permuting \$A\$, \$B\$, \$C\$ if necessary, we may take \$D\$ odd and not divisible by 5, and \$C < D\$ such that \$D^4 - C^4\$ is divisible by 625 and satisfies several other congruence and divisibility properties, and for each such \$D\$ and \$C\$ look for a representation of \$D^4 — C^4\$ as \$A^4 + B^4\$ with \$A\$, \$B\$ divisible by 5. Frye translated this into a computer program and ran it on various Connection Machines for about 100 hours to find the minimal counterexample to Euler's conjecture: $$958004 + 2175194 + 4145604 = 4224814.$$ He continued the search and found that this solution is unique in the range \$D < 10^6\$. This solution appears on the parameterization (6) with \$(m, n) = (20,-9)\$. We include Frye's result with his permission.

Postscript.

$$(A, B, C;D) = (2682440,15365639,18796760; 20615673)$$ to Euler's conjecture still seems beyond the range of reasonable exhaustive computer search, there remained the possibility that smaller solutions may be found by such a search. Shortly after hearing of the first solution, Roger Frye of Thinking Machines Corporation asked whether it was minimal; I did not know, but suggested how one might exhaustively search for smaller solutions: eliminating common factors and permuting \$A\$, \$B\$, \$C\$ if necessary, we may take \$D\$ odd and not divisible by 5, and \$C < D\$ such that \$D^4 - C^4\$ is divisible by 625 and satisfies several other congruence and divisibility properties, and for each such \$D\$ and \$C\$ look for a representation of \$D^4 — C^4\$ as \$A^4 + B^4\$ with \$A\$, \$B\$ divisible by 5. Frye translated this into a computer program and ran it on various Connection Machines for about 100 hours to find the minimal counterexample to Euler's conjecture: $$95800^4 + 217519^4 + 414560^4 = 422481^4.$$ He continued the search and found that this solution is unique in the range \$D < 10^6\$. This solution appears on the parameterization (6) with \$(m, n) = (20,-9)\$. We include Frye's result with his permission.