# Why don't I get numeric values from this Solve?

Solve[ a^2 + b^2 == c^2 && a < 100 && b < 100 && c < 100, {a, b, c}, Integers]


gives me this result

{{c -> ConditionalExpression[-Sqrt[ a^2 + b^2],
((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] - b > 0  && -99 <= a <= -1) ||
((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] - b > 0  &&   1 <= a <= 99) ||
((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] + b <= 0 && -99 <= a <= -1) ||
((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] + b <= 0 &&   1 <= a <= 99) ||
...           ] },   ...  }


where I would expect something like

{{x -> 3, y -> 4, z -> 5}, {x -> 4, y -> 3, z -> 5}, {x -> 5, y -> 12, z -> 13},...}


What am I doing wrong?

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Your first 1000 of rep! That many more may follow. – Sjoerd C. de Vries Sep 10 '12 at 5:19
Yeah! congrats! – Dr. belisarius Sep 10 '12 at 5:20
@Sjoerd - Whohoo! Party time! :-) Still a long way to 71k I'm afraid :-) – stevenvh Sep 10 '12 at 5:24
@stevenvh grats! wow, I did not realize you were leading on electronics - nice score ;) – Vitaliy Kaurov Sep 10 '12 at 15:48
@Vitaliy - Thanks. Hey, I'm doing great: another 5 rep, already at 1045!, so 70k by Thanksgiving may be possible after all :-). I'll have to do a lot of studying on Mathematica until then, I'm afraid. – stevenvh Sep 10 '12 at 16:12

This works (it's obvious the same will hold for negative integers):

Reduce[a^2 + b^2 == c^2 && 0 < a < 100 && 0 < b < 100 && 0 < c < 100, {a, b, c}, Integers]


And if you're up for an interesting visualization, here is my take:

ListPlot3D[{a, b, c} /. Solve[a^2 + b^2 == c^2 && 0 < a < 100 && 0 < b < 100 &&
0 < c < 100, {a, b, c}, Integers], InterpolationOrder -> 0,
Mesh -> None, ColorFunction -> "Rainbow", Filling -> Bottom, BoxRatios -> 1]


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Thanks. Also seems to work with Solve if I include the lower limits of 0. Any reason why I should use Reduce instead of Solve, other than the different output format? They seem equally fast. – stevenvh Sep 10 '12 at 5:22
@stevenvh You may find the link at the end of my answer useful for the question in your comment above – Dr. belisarius Sep 10 '12 at 5:50
I can't upvote again, but I like the scaly plot... looks like some exotic fish skeleton. :) – J. M. Sep 10 '12 at 16:31
That plot needs to be on some superhero's chest... – R. M. Sep 10 '12 at 18:00
For c given the larger number of solutions the larger grains should be, but they aren't. – Artes Sep 10 '12 at 18:07

The result we get is correct and one shouldn't expect it in a different form since there are infinitely many solutions and there is no way to enumerate them all. That's why Mathematica returns the result in a symbolic way. This issue appears a bit more obviously when one looks for all solutions in special subclasses, e.g. solutions of this form :

a == 3k, b == 4k, c == 5k for k < 20 && k ∈ Integers, e.g. k == -10^100000. We have to use Reduce instead of Solve in this case :

Reduce[(3 k)^2 + (4 k)^2 == (5 k)^2 && k < 20, k, Integers]

k ∈ Integers && k <= 19


There are infinitely many solutions, however if we add a lower bound for k we'll get a finite numbers of explicit solutions :

Reduce[(3 k)^2 + (4 k)^2 == (5 k)^2 && 15 < k < 20, k, Integers]

k == 16 || k == 17 || k == 18 || k == 19


Let's visualize an interesting part of the solution space to the general problem :

pts = Solve[{ a^2 + b^2 == c^2,
-300 < a < 100, -300 < b < 100, -300 < c < 100}, {a, b, c}, Integers];
Length[pts]

2345

Show[ ContourPlot3D[ x^2 + y^2 == z^2, {x, -300, 100}, {y, -300, 100}, {z, -300, 100},
ContourStyle -> Directive[Orange, Opacity[0.3], Specularity[White, 20]],
MeshFunctions -> {#3 &}, Mesh -> 7, PlotPoints -> 100],
ListPointPlot3D[ {a, b, c} /. pts, PlotStyle -> Directive[Darker@Red, PointSize[0.004]]]]


These cones (the upper and a part of the lower) denote all real solutions whereas the red points denote all integer sloutions of the sytem : { a^2 + b^2 == c^2, -300 < a < 100, -300 < b < 100, -300 < c < 100}

One might be also interested in integer solutions modulo 100 to find all solutions of this type : a^2 + b^2 == c^2 + 100 k for k ∈ Integers (i.e. solutions satisfy 0 <= a < 100, 0 <= b < 100, 0 <= c < 100 ), there are many more such solutions ( i.e. 17400) :

Solve[ a^2 + b^2 == c^2, {a, b, c}, Modulus -> 100] // Short[#, 2] &

{{a -> 0, b -> 0, c -> 0},  {a -> 0, b -> 0, c -> 10}, {a -> 0, b -> 0, c -> 20},
{a -> 0, b -> 0, c -> 30},   <<17393>>,     {a -> 99, b -> 90, c -> 49},
{a -> 99, b -> 90, c -> 51}, {a -> 99, b -> 90, c -> 99}}

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Your cone is too steep. Some points are falling off :) – Dr. belisarius Sep 10 '12 at 15:35
@belisarius The solution of this problem is out of the scope of the question :) – Artes Sep 10 '12 at 15:39
@belisarius Nevertheless I've corrected the plot. – Artes Sep 12 '12 at 22:49
Ok. I will remove my downvote ;) – Dr. belisarius Sep 12 '12 at 22:51

Bound the variables from both sides if you want to get explicit results

s = Solve[a^2+b^2==c^2&& -100 < a < 100&& -100 < b < 100&& -100 < c < 100, {a,b,c}, Integers]
(* -> {{a -> -99, b -> 0, c -> -99}, {a -> -98, b -> 0, c -> 98} ... *)
ListPointPlot3D[{a, b, c} /. s]


Or you could reduce the domain to positive integers, taken advantage of the problem being quadratic on all vars:

s = Solve[a^2 + b^2 - c^2 == 0 &&
0 <= a < 100 && 0 <= b < 100 &&  0 <= c < 100, {a, b, c}, Integers]

ListPointPlot3D[{a, b, c} /. s]


Edit

And you can verify that

2 Sum[SquaresR[2, i^2], {i, 0, 99}] -1 == (2 Length@# -Count[#, 0]) &@Flatten[{a, b, c} /. s]
(* True *)


This is a useful reading when solving Diophantine equations

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