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I am trying to solve the equation $(3^m - 3^n)^2 = 2^a + 2^b$ where $m, n, a, b$ be positive integers. I tried

Solve[{(3^m - 3^n)^2 == 2^a + 2^b, 1 <= n <= 10, 1 <= m <= 10, 
  1 <= a <= 20, 1 <= b <= 20}, {m, n, a, b}, Integers]

I can not get any solution. How can I solve it? I know that, $m=3$, $n=1$, $a=9$, and $b=6$ is a solution of the equation.

f[m_, n_, a_, b_] = (3^m - 3^n)^2 - 2^a - 2^b
f[3, 1, 9, 6]

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    $\begingroup$ f[{m_,n_,a_,b_}]:=(3^m-3^n)^2==2^a+2^b;Select[Tuples[Range[1,9],4],f] $\endgroup$
    – Bill
    Commented Dec 2, 2023 at 4:20
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    $\begingroup$ Using f as defined by @Bill, Select[Tuples[ {Range[10], Range[10], Range[20], Range[20]}], f] Gives same result but checked full range of variables. $\endgroup$
    – Bob Hanlon
    Commented Dec 2, 2023 at 7:05

1 Answer 1

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The following succeeds:

NMinimize[{((3^m - 3^n)^2 - (2^a + 2^b))^2, 1 <= n <= 10, 
1 <= m <= 10, 1 <= a <= 20, 1 <= b <= 20, 
{m, n, a, b} \[Element] PositiveIntegers}, {m, n, a, b}]

{1.29247*10^-26, {m -> 1, n -> 3, a -> 9, b -> 6}}

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