Can you find two integers m, n such that 2^m+n= 2^mn?

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Solution

Let m = 1, n = 1

⇒ m + n = 1 + 1 = 2 and mn = 1 × 1 = 1

⇒ m + n ≠ mn

Let m = 1, n = 2

⇒ m + n = 1 + 2 = 3 and mn = 1 × 2 = 2

⇒ m + n ≠ mn

Let m = 2, n = 1

⇒ m + n = 2 + 1 = 3 and mn = 2 × 1 = 2

⇒ m + n ≠ mn

Let m = 2, n = 2

⇒ m + n = 2 + 2 = 4 and mn = 2 × 2 = 4

⇒ m + n = mn

∴ m = n = 2

⇒

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