Inspired by Adebayo Iyanu

Number Theory Level 4

n x = n 2 x \Large n^x=n^2x

How many ordered pairs ( n , x ) (n,x) are there satisfying above equation for n , x ∈ Z − { 0 } n,x \in \mathbb{Z}-\{0\} ?

Inspiration

5 15 Infinitely many 4 9 11

Nihar Mahajan by Nihar Mahajan , 20, India

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1 solution

Garrett Clarke
Jul 31, 2015

Not a proof, but here are the solutions! In the form (n,x):

(-2,4), (-1,-1), (1,1), (2,4) and (3,3)

I got my answers by rearranging the equation to n = x 1 x − 2 n=x^{\frac{1}{x-2}} and checking for solutions by hand. It's apparent that for larger ∣ x ∣ |x| , 1 x − 2 \frac{1}{x-2} will be too small for n n to be an integer.

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