Why does Solve lock up when trying to solve the quadratic equation with large integers?












4












$begingroup$


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.






equation-solving







share|improve this question









New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • 1




    $begingroup$
    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
    4 hours ago






  • 1




    $begingroup$
    I quickly get solutions for 2^257 (V11.3.0, macos).
    $endgroup$
    – Michael E2
    4 hours ago










  • $begingroup$
    Sorry, there was a typo - it should be 2^511
    $endgroup$
    – user63373
    4 hours ago






  • 1




    $begingroup$
    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
    3 hours ago








  • 2




    $begingroup$
    @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
    3 hours ago
















4












$begingroup$


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.






equation-solving







share|improve this question









New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    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
    4 hours ago






  • 1




    $begingroup$
    I quickly get solutions for 2^257 (V11.3.0, macos).
    $endgroup$
    – Michael E2
    4 hours ago










  • $begingroup$
    Sorry, there was a typo - it should be 2^511
    $endgroup$
    – user63373
    4 hours ago






  • 1




    $begingroup$
    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
    3 hours ago








  • 2




    $begingroup$
    @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
    3 hours ago














4












4








4





$begingroup$


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.






equation-solving







share|improve this question









New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




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.






equation-solving




equation-solving






share|improve this question









New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|improve this question









New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|improve this question




share|improve this question








edited 2 hours ago









mikado

6,6471929




6,6471929






New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 5 hours ago









user63373user63373

233




233




New contributor




user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






user63373 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    $begingroup$
    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
    4 hours ago






  • 1




    $begingroup$
    I quickly get solutions for 2^257 (V11.3.0, macos).
    $endgroup$
    – Michael E2
    4 hours ago










  • $begingroup$
    Sorry, there was a typo - it should be 2^511
    $endgroup$
    – user63373
    4 hours ago






  • 1




    $begingroup$
    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
    3 hours ago








  • 2




    $begingroup$
    @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
    3 hours ago














  • 1




    $begingroup$
    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
    4 hours ago






  • 1




    $begingroup$
    I quickly get solutions for 2^257 (V11.3.0, macos).
    $endgroup$
    – Michael E2
    4 hours ago










  • $begingroup$
    Sorry, there was a typo - it should be 2^511
    $endgroup$
    – user63373
    4 hours ago






  • 1




    $begingroup$
    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
    3 hours ago








  • 2




    $begingroup$
    @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
    3 hours ago








1




1




$begingroup$
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
4 hours ago




$begingroup$
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
4 hours ago




1




1




$begingroup$
I quickly get solutions for 2^257 (V11.3.0, macos).
$endgroup$
– Michael E2
4 hours ago




$begingroup$
I quickly get solutions for 2^257 (V11.3.0, macos).
$endgroup$
– Michael E2
4 hours ago












$begingroup$
Sorry, there was a typo - it should be 2^511
$endgroup$
– user63373
4 hours ago




$begingroup$
Sorry, there was a typo - it should be 2^511
$endgroup$
– user63373
4 hours ago




1




1




$begingroup$
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
3 hours ago






$begingroup$
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
3 hours ago






2




2




$begingroup$
@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
3 hours ago




$begingroup$
@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
3 hours ago










1 Answer
1






active

oldest

votes


















6












$begingroup$

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.






share|improve this answer









$endgroup$













  • $begingroup$
    Thank you very much. This seems to be exactly what is happening. Much appreciated.
    $endgroup$
    – user63373
    2 hours ago










  • $begingroup$
    @user63373 You're welcome. :)
    $endgroup$
    – Michael E2
    1 hour ago











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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









6












$begingroup$

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.






share|improve this answer









$endgroup$













  • $begingroup$
    Thank you very much. This seems to be exactly what is happening. Much appreciated.
    $endgroup$
    – user63373
    2 hours ago










  • $begingroup$
    @user63373 You're welcome. :)
    $endgroup$
    – Michael E2
    1 hour ago
















6












$begingroup$

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.






share|improve this answer









$endgroup$













  • $begingroup$
    Thank you very much. This seems to be exactly what is happening. Much appreciated.
    $endgroup$
    – user63373
    2 hours ago










  • $begingroup$
    @user63373 You're welcome. :)
    $endgroup$
    – Michael E2
    1 hour ago














6












6








6





$begingroup$

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.






share|improve this answer









$endgroup$



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.







share|improve this answer












share|improve this answer



share|improve this answer










answered 2 hours ago









Michael E2Michael E2

148k12198478




148k12198478












  • $begingroup$
    Thank you very much. This seems to be exactly what is happening. Much appreciated.
    $endgroup$
    – user63373
    2 hours ago










  • $begingroup$
    @user63373 You're welcome. :)
    $endgroup$
    – Michael E2
    1 hour ago


















  • $begingroup$
    Thank you very much. This seems to be exactly what is happening. Much appreciated.
    $endgroup$
    – user63373
    2 hours ago










  • $begingroup$
    @user63373 You're welcome. :)
    $endgroup$
    – Michael E2
    1 hour ago
















$begingroup$
Thank you very much. This seems to be exactly what is happening. Much appreciated.
$endgroup$
– user63373
2 hours ago




$begingroup$
Thank you very much. This seems to be exactly what is happening. Much appreciated.
$endgroup$
– user63373
2 hours ago












$begingroup$
@user63373 You're welcome. :)
$endgroup$
– Michael E2
1 hour ago




$begingroup$
@user63373 You're welcome. :)
$endgroup$
– Michael E2
1 hour ago










user63373 is a new contributor. Be nice, and check out our Code of Conduct.










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南乌拉尔铁路局

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南乌拉尔铁路局 Южно-Уральская железная дорога 成立 1934年1月4日 国家   蘇聯（1934年—1991年）   俄羅斯（1991年—） 总部 车里雅宾斯克 隶属 俄罗斯铁路股份公司 路局代码 80 铁路里程 4,934.9公里 奖项 网址 http://yuzd.rzd.ru/ 查 论 编 南乌拉尔铁路局 （俄语： Южно-Уральская железная дорога ， 羅馬化： Yuzhno-Uralskaya zheleznaya doroga ，简称 ЮУЖД ）是俄罗斯铁路股份公司属下的铁路管理局之一，总部位于车里雅宾斯克，负责管辖俄罗斯联邦乌拉尔南部地区的铁路系统，范围包括奥伦堡州、车里雅宾斯克州、库尔干州的大部分铁路，以及萨马拉州、斯维尔德洛夫斯克州、巴什科尔托斯坦共和国的少部分铁路，并有部分铁路延伸至或穿过哈萨克斯坦境内。南乌拉尔铁路局的营业里程为4,934.9公里，与伏尔加铁路局、古比雪夫铁路局、斯维尔德洛夫斯克铁路局、西西伯利亚铁路局相邻，并与南方的哈萨克斯坦铁路连接。 外部链接 （俄文） Южно-Уральская железная дорога — филиал ОАО РЖД （俄文） 南乌拉尔铁路局管辖路线图（一） （俄文） 南乌拉尔铁路局管辖路线图（二） 查 论 编 俄罗斯铁路（集团）股份公司 十月铁路局 · 莫斯科铁路局 · 北高加索铁路局 · 高尔基铁路局 · 古比雪夫铁路局 · 北方铁路局 · 伏尔加铁路局 · 东南铁路局 · 斯维尔德洛夫斯克铁路局 · 南乌拉尔铁路局 · 西西伯利亚铁路局 · 克拉斯诺亚尔斯克铁路局 · 东西伯利亚铁路局 · 后贝加尔铁路局 · 远东铁路局 · 加里宁格勒铁路局 This page is only for reference, If you need detailed information, please check here
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