Congruent Number

Author

Eric W. Weisstein
November 20, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DiophantineEquations/CongruentNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CongruentNumber.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Special Case a==101

Reduce[x^2+a y^2==z^2&&x^2-a y^2==t^2&&x>0&&y>0&&z>0&&t>0&&a==101,
{x,y,z,t},Integers]

?FindInstance

With[{a=101},
FindInstance[x^2+a y^2==z^2&&x^2-a y^2==t^2&&x>0&&y>0&&z>0&&t>0,{x,y,z,t},Integers]
]

Developer`SetSystemOptions["ReduceOptions"->{"SieveMaxPoints"->10^6}];
With[{a=101},
FindInstance[x^2+a y^2==z^2&&x^2-a y^2==t^2&&x>0&&y>0&&z>0&&t>0,{x,y,z,t},Integers]
]//Timing

Developer`SetSystemOptions["ReduceOptions"->{"SieveMaxPoints"->10^7}];
With[{a=101},
FindInstance[x^2+a y^2==z^2&&x^2-a y^2==t^2&&x>0&&y>0&&z>0&&t>0,{x,y,z,t},Integers]
]//Timing

With[{a=101},
x^2+a y^2==z^2&&x^2-a y^2==t^2
]/.{x->2015242462949760001961,y->118171431852779451900,z->2339148435306225006961,t->1628124370727269996961}