Solving Diophantine equation warning: not all solutions

Questions
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1

Based on the description here:

Selection_247.png
Selection_247.png1598Ă—724 186 KB

I tried to solve the following Diophantine equation but no specific solution has been obtained:

In[172]:= Solve[ x^4 + y^4 + z^4 == w^4, {x,y,z,w},Integers]

During evaluation of In[172]:= Solve::svars: Equations may not give solutions for all "solve" variables.

Out[172]= {{w -> 
ConditionalExpression[-(x^4 + y^4 + z^4)^(
1/4), (x | y | z | (x^4 + y^4 + z^4)^(1/4)) \[Element] 
Integers]}, {w -> 
ConditionalExpression[(x^4 + y^4 + z^4)^(
1/4), (x | y | z | (x^4 + y^4 + z^4)^(1/4)) \[Element] Integers]}}

I want to know how to get the results shown in the Wikipedia above.

Regards,
Zhao

2

I do not know how to give you the Wiki entry, but perhaps this will help you

 FindInstance[ x^4 + y^4 + z^4 == w^4&&x>0&&y>0&&z>0, {x,y,z,w},Integers]
3

Thank you for pointing out this trick. The following methods are equivalent:

In[9]:= FindInstance[ x^4 + y^4 + z^4 == w^4 &&x>0&&y>0&&z>0, {x,y,z,w},Integers]
FindInstance[ x^4 + y^4 + z^4 == w^4, {x,y,z,w},PositiveIntegers]

Out[9]= {{x -> 18796760, y -> 2682440, z -> 15365639, w -> 20615673}}

Out[10]= {{x -> 18796760, y -> 2682440, z -> 15365639, w -> 20615673}}

Furthermore, I also noticed that in order to obtain the same result as the Wiki entry, at least two variables need to be set to known values:

In[33]:= FindInstance[ {x^4 + y^4 + z^4 == w^4,  x==95800 }, {x,y,z,w},PositiveIntegers]
FindInstance[ {x^4 + y^4 + z^4 == w^4,  x==95800,y==217519 }, {x,y,z,w},PositiveIntegers]

During evaluation of In[33]:= FindInstance::nsmet: The methods available to FindInstance are insufficient to find the requested instances or prove they do not exist.

Out[33]= FindInstance[{x^4 + y^4 + z^4 == w^4, x == 95800}, {x, y, z, 
  w}, PositiveIntegers]

Out[34]= {{x -> 95800, y -> 217519, z -> 414560, w -> 422481}}

Nevertheless, the following method still fails to do the trick:

In[44]:= FindInstance[ {x^4 + y^4 + z^4 == w^4,  x==95800, w == 422481}, {x,y,z,w},PositiveIntegers]

Out[44]= $Aborted
4

I acknowledge that some, perhaps many, users want exactly the solution that they want with exactly the fonts and published form that they want and to get this in a few seconds.

I want less.

I think it is astonishing that without any help it correctly finds any solution at all in a few seconds. I did not expect that it would be able to do this. I cannot imagine how this is possible.

You can ask FindInstance to search for multiple solutions, but I don’t think in this case that will succeed in finding your desired solution without additional help.

I would like to know how many computer years it took Frye to find his solution. Since he published his result in “Proceedings of Supercomputing” I expect it was a stunning amount of computer time and power if compared to the tiny amount of computing power that I have.

5

Thank you for pointing this out. I attached the original paper by Roger E.Frye for reference.

superc.1988.74138.pdf (530 KB)