Find the smallest integer **N** which satifies:

I did not specify the domains of N, so we could have a discussion and consider several cases

**Case #1**

N is a positive integer, and a, b integers.

The least value of N is 1. This occurs for (a, b) = (-2, 1)

**Case #2** N, a, b positive integers

Let b = k*a, with rational k > 1, then

It is clear that a and b are integers if and only if both k and 3/(k-l) are integers,

that is. if and only if k = 2 or k = 4.

k = 2 —> (a, b) = (16, 32) —>

and

k = 4 —> (a, b) = (16, 64) —>

The smallest integer is

Now consider **case #3** N positive integer and a, b positive real numbers.

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If we allow negative values for a or and b the smallest value of N is 1, if not then it is quite large. Format is {a, b, N}

{-2, 1, 1}

{16,32,2135987035920910082395021706169552114602704522356652769947041607822219725780640550022962086936576}.

I’m not 100% but I recon these are the only solutions for a and b when N is an integer.

{-4,32}

{-2,1}

{16,32}

{16,64}

There are a few more where a^(a+2b) = b^(b+2a) but N is fractional.

Paul.

Please read my post and case #3