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Why does Solve lock up when trying to solve the equation

Solve[(x^2+y^2)+(x+y)==2^511 && x>0 && y>0,{x,y},Integers]

It works up 2^185, but at higher powers of 2, it seems to stop processing. The program says it's running, but there is no solution after running overnight. Running Mathematica 11.3 on Windows 32-bit OS.

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    $\begingroup$ You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is useful for learning how to format your questions and answers. You may also find this meta Q&A helpful $\endgroup$ – Michael E2 Mar 10 at 18:27
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    $\begingroup$ I quickly get solutions for 2^257 (V11.3.0, macos). $\endgroup$ – Michael E2 Mar 10 at 18:29
  • $\begingroup$ Sorry, there was a typo - it should be 2^511 $\endgroup$ – user63373 Mar 10 at 19:21
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    $\begingroup$ Probably it's combinatorial blowup. n = 185 gives 32 solutions but n = 257 already gives 1024. Might be more work and memory to store the symbolics than your CPU can handle. $\endgroup$ – b3m2a1 Mar 10 at 19:38
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    $\begingroup$ @b3m2a1 I think the reasons probably have to do with number theory. n = 323 produces 8192 solutions in 2.3s and n = 325 produces only 128 solutions in 500s. The memory growth is quite low. I think for n = 511, you just have to wait long enough, and I can't predict how long that is. $\endgroup$ – Michael E2 Mar 10 at 19:57
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Here's a guess: The Diophantine problem $$ x^2+y^2+x+y=a$$ is equivalent, via $u=2x+1,v=2y+1$ to finding the odd solutions to $$u^2+v^2=2+4a \,.$$ Whether Solve makes this transformation or not, solving the Pythagorean equation can be done from the prime factorization of $2+4a$. How long Solve takes thus might depend on how long it takes to factor $2+4a$.

This is not hard to verify:

Block[{FactorInteger = (Print["FactorInteger"[##]]; Abort[]) &},
 PrintTemporary@Dynamic@Clock@Infinity;
 Print[2 + 4 2^325];
 Solve[(x^2 + y^2) + (x + y) == 2^325 && x > 0 && y > 0, {x, y}, Integers] // AbsoluteTiming
 ]
  2734063405978764905465627783897026706691461788616515545532213258012441248999219\
   90402939147127881730

FactorInteger[
  2734063405978764905465627783897026706691461788616515545532213258012441248999219\
   90402939147127881730]

$Aborted

Well, it turns out it takes about 500 sec. to factor 2 + 4 * 2^325, which is about how long it takes the Solve above to run.

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  • $\begingroup$ Thank you very much. This seems to be exactly what is happening. Much appreciated. $\endgroup$ – user63373 Mar 10 at 21:19
  • $\begingroup$ @user63373 You're welcome. :) $\endgroup$ – Michael E2 Mar 10 at 22:10

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