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Solve this equation using Mathematica:

$2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.

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  • $\begingroup$ Exponential Diophantine equations are difficult in general. That being said, look at Reduce[] and FindInstance[]. $\endgroup$ Commented Sep 8, 2017 at 17:07
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    $\begingroup$ Isn't this two equations in three unknowns? $\endgroup$ Commented Sep 8, 2017 at 20:39

2 Answers 2

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Dealing with diophantine equations after appropriate restriction of the possible solution space one could play with extension of ExhaustiveSearchMaxPoints. 

SystemOptions["ReduceOptions" -> "ExhaustiveSearchMaxPoints"]

For an example when an extension appears crucial see e.g. Solving/Reducing equations in Z/pZ. Here I just set (I don't insist it is neccessary) :

SetSystemOptions["ReduceOptions" -> "ExhaustiveSearchMaxPoints" -> {1000, 1000000}];

and playing with Reduce I just found after roughly two minutes

Reduce[ 2^x + y^113 == z^2 && GCD[x, y] == 1 && 500 >= x > 0 && 32 >= y > 0 && z > 0, {x, y, z}, Integers] 
 (x == 3 && y == 1 && z == 3) || (x == 113 && y == 2 && z == 144115188075855872) || (x == 229 && y == 4 && z == 31153781151208965771182977975320576) || (x == 234 && y == 1 && z == 166153499473114484112975882535043072) || (x == 236 && y == 1 && z == 332306998946228968225951765070086144) || (x == 238 && y == 1 && z == 664613997892457936451903530140172288) || (x == 240 && y == 1 && z == 1329227995784915872903807060280344576) || (x == 242 && y == 1 && z == 2658455991569831745807614120560689152) || (x == 339 && y == 8 && z == 1496577676626844588240573268701473812127674924007424) || (x == 455 && y == 16 && z == 32351936000580767753600418104423556808364573307048686977324332\ 2990592)

This result suggest that the only solutions are of the form y == 2^n for natural n. In fact, I've looked for solutions with higher bounds for x and y and all solutions were of the form y == 2^n. It seems to be easy to prove, nonetheless I put aside this step up to you. This can significantly simplify the search for solutions. Moreover it might be quite resonable to look for z expressed by its prime components. Now we can provide more solutions (it took a few minutes)

sols = {x, n, z} /. { ToRules @ Reduce[2^x + (2^n)^113 == z^2 && GCD[x, 2^n] == 1 && 2000 >= x > 0 && 20 >= n >= 0 && z > 0, {x, n, z}, Integers]}; {#1, #2, FactorInteger @ #3} & @@@ sols // Column 
{3, 0, {{3, 1}}} {113, 1, {{2, 57}}} {229, 2, {{2, 113}, {3, 1}}} {234, 0, {{2, 117}}} {236, 0, {{2, 118}}} {238, 0, {{2, 119}}} {240, 0, {{2, 120}}} {242, 0, {{2, 121}}} {339, 3, {{2, 170}}} {455, 4, {{2, 226}, {3, 1}}} {565, 5, {{2, 283}}} {681, 6, {{2, 339}, {3, 1}}} {791, 7, {{2, 396}}} {907, 8, {{2, 452}, {3, 1}}} {1017, 9, {{2, 509}}} {1133, 10, {{2, 565}, {3, 1}}} {1243, 11, {{2, 622}}} {1359, 12, {{2, 678}, {3, 1}}} {1469, 13, {{2, 735}}} {1585, 14, {{2, 791}, {3, 1}}} {1695, 15, {{2, 848}}} {1811, 16, {{2, 904}, {3, 1}}} {1921, 17, {{2, 961}}}

These are the only solutions under the above bounds. With this approach it is quite natural to generalize and find all possible solutions proving a simple theorem. At this point it is reasonable to stop since you haven't provided any piece of code.

Edit

It appears that the above analysis was performed for a different equation than the original one (see a comment by anderstood) i.e. we should do
something like this (it took more than 15 minutes to evaluate):

 Reduce[ 2^x + 113^y == z^2 && GCD[x, y] == 1 && 2000 >= x > 0 && 100 >= y > 0 && z > 0, {x, y, z}, Integers] 
(x == 3 && y == 1 && z == 11) || (x == 9 && y == 1 && z == 25) || (x == 242 && y == 1 && z == 2658455991569831745807614120560689152)

One can easily check that the first 2 solutions are correct however
there is an awful bug in the system behind the third solution. Namely z == 2658455991569831745807614120560689152 == 2^121 therefore 2^242 cannot be equal to 113 + (2^121)^2.

All of the following produce the same flawed results (in Mathematica versions 10.1, 10.3, 11.1)

Reduce[2^242 + 113 == z^2 && z > 0, {z}, Integers] Reduce[2^242 + 113^1 == z^2 && z ∈ Integers && z > 0, z] Solve[z^2 == 2^242 + 113 && z ∈ Integers && z > 0, z]

while 

Solve[2^242 + 113^1 == z^2 && z > 0, z, Integers]

does not.

I'm going to leave this post as a warning sign (until the bug is present) for the comunity, although we've got only partial results.

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    $\begingroup$ Isn't it suppose to be $113^y$ instead of $y^{113}$? Then $y=2^n$ no longer holds but your answer can be easily adapted and works. $\endgroup$ Commented Sep 8, 2017 at 20:46
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    $\begingroup$ With Reduce[2^x + 113^y == z^2 && GCD[x, y] == 1 && 2000 >= x > 0 && 100 >= y > 0 && z > 0, {x, y, z}, Integers] I got (x == 3 && y == 1 && z == 11) || (x == 9 && y == 1 && z == 25) || (x == 242 && y == 1 && z == 2658455991569831745807614120560689152). $\endgroup$ Commented Sep 8, 2017 at 20:57
  • $\begingroup$ @anderstood I've found a bug in the system, and can't proceed further now. You can see that the last triple (x == 242 && y == 1 && z == 2658455991569831745807614120560689152) in your comment cannot be a solution since z==2^242. The problem lies in Reduce and partially also in Solve. $\endgroup$ Commented Sep 9, 2017 at 16:04
  • $\begingroup$ Do you think it could be due to a truncation error (I mean that $2^{242}$ is so large that $113$ is seen as $0$). $\endgroup$ Commented Sep 9, 2017 at 18:17
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    $\begingroup$ @anderstood No, compare just Solve[z^2 == 2^242 + 113 && z ∈ Integers && z > 0, z] and Solve[2^242 + 113^1 == z^2 && z > 0, z, Integers] both should provide the same results (no solutions) however the latter yields incorrect one. For exact arithmetic there are no approximations. $\endgroup$ Commented Sep 9, 2017 at 18:23
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Since Reduce and Solve don't work for higher values, simpliy test it directly, which goes quite fast.

First make a ContourPlot to get an impression, where solutions are to be expected.

ContourPlot3D[2^x + 113^y == z^2, {x, 0, 210}, {y, 0, 30}, {z, 0, 10^30}]

enter image description here

Solutions for given y are up to about x = 7*y and z = 10^y

Now do direct testing with y up to 100, x up to 700 and z up to very high z = Sqrt[2^700+113^100] = 2.2935*10^105

(sol = Flatten[ Table[If[ GCD[x, y] == 1 && IntegerQ[z = Sqrt[2^x + 113^y]], {{x, y, z}}, {}], {x, 0, 700}, {y, 0, 100}], 2]) // Timing

Quite fast you get the two solutions

(* {62.641, {{3, 1, 11}, {9, 1, 25}}} *)
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  • $\begingroup$ thank you everone. I want solutions only in integers and it is possible to give a general solution of this equation. $\endgroup$ Commented Sep 13, 2017 at 16:52

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