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Computer Science Mathematics Mathematica Algebra Equation Solving Wolfram Language Optimization Numerical Computation

Improving calculation speed for finding perfect squares using two variables

Jan Eerland

Posted 4 years ago

Well, I have the following code: ParallelTable[ If[TrueQ[ Sqrt[123(3(2x)^2+167y^2x+yx+4y^3x^2)] \[Element] Integers], {{x, y}, Nothing}], {x, 2000,5000}, {y, 1000000000, 1000001000}] //. Null -> Nothing //. {} -> Nothing And using this code I am trying to determine for what $(x,y)$ the function $123(3(2x)^2+167y^2x+yx+4y^3x^2)$ gives a perfect square. Is there a way to improve the speed of this calculation? Every $1000$ possibilities take now round about 14 minutes, which is a bit slow. Is there a way to check if the function I stated in the question is a perfect square, in a more sophisticated way?

POSTED BY: Jan Eerland

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Rohit Namjoshi

Posted 4 years ago

Quick comparison Simplify[123(3(2x)^2 + 167y^2x + yx + 4y^3x^2)] (* 123 x (y (1 + 167 y) + 4 x (3 + y^3) ) f[x_, y_] := 123 x (y (1 + 167 y) + 4 x (3 + y^3); Table[If[TrueQ[Sqrt[f[x, y]] \[Element] Integers], {x, y}, Nothing], {x, 2000, 5000}, {y, 2000, 5000}] // AbsoluteTiming // First ( I gave up after waiting for 6 minutes ) Table[If[sQ[f[x, y]], {x, y}, Nothing], {x, 2000, 5000}, {y, 2000, 5000}] // AbsoluteTiming // First ( 49.3531 *)

Quick comparison

Simplify[123*(3*(2*x)^2 + 167*y^2*x + y*x + 4*y^3*x^2)]
(* 123 x (y (1 + 167 y) + 4 x (3 + y^3) *)

f[x_, y_] := 123 x (y (1 + 167 y) + 4 x (3 + y^3);

Table[If[TrueQ[Sqrt[f[x, y]] \[Element] Integers], {x, y}, Nothing], {x, 2000, 5000},
  {y, 2000, 5000}] // AbsoluteTiming // First
(* I gave up after waiting for 6 minutes *)

Table[If[sQ[f[x, y]], {x, y}, Nothing], {x, 2000, 5000}, {y, 2000, 5000}] // 
 AbsoluteTiming // First
(* 49.3531 *)

POSTED BY: Rohit Namjoshi