Diophantine Equation—4th Powers

Author

Eric W. Weisstein
March 8, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DiophantineEquations/DiophantineEquation4thPowers.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DiophantineEquation4thPowers.html.

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

Like Powers

And@@Equal@@@Map[Plus@@#&,LikePowers[4,1,4]^4,{2}]

Equal@@@({Plus@@First[#],#[[-1,-1]]}&/@(Map[HoldForm,Reverse/@LikePowers[4,1,4],{-1}]^4))//ColumnForm//TraditionalForm

Kyrtatas's Equation

(((a^4)+(b^4)+(c^4))/((d^4)+(e^4)+(f^4)))==((a+b+c)/(d+e+f))/.Thread[{a,b,c,d,e,f}->#]&/@{
{3,25,38,7,20,39},
{285,2964,3249,1769,1952,3721},
{185,1184,1369,663,858,1521},
(* Stinchcomb, Nov. 19, 2004: *)
{5160,11481,16641,3683,12446,16129},  {7367,11954,19321,2660,15029,17689},
{14925,24676,39601,7527,29722,37249},
{7136,42593,49729,2387,44702,47089}
}

Reduce[(((a^4)+(b^4)+(c^4))/((d^4)+(e^4)+(f^4)))==((a+b+c)/(d+e+f)),{a,b,c,d,e,f},Integers]//Timing

vars={a,b,c,d,e,f};
TimeConstrained[FindInstance[(((a^4)+(b^4)+(c^4))/((d^4)+(e^4)+(f^4)))==((a+b+c)/(d+e+f))&&And@@(#>1&/@vars),vars,Integers],120]//Timing

Piezas

eqn=Subtract@@{(a x^2+2(b+c)x y-3a y^2)^k+(b x^2-2(a+c)x y-3b y^2)^k+(c x^2-2(a-b)x y-3c y^2)^k,(a^k+b^k+c^k)(x^2+3y^2)^k};

Expand[eqn/.{c->a+b,k->2}]

Expand[eqn/.{c->a+b,k->4}]

Expand[Subtract@@{(4x^2+12y^2)^4+(15x^2+45y^2)^4+(50x^2+300x y-150y^2)^4+(50x^2-300x y-150y^2)^4+(100x^2-300y^2)^4,(103x^2+309y^2)^4}]