How to find an integer solution to a nonlinear equation?

[al4n]

given something like: aⁿ = c
where c is given and a, n, and c are only allowed to be integers. how would one find the value of say n or a?

 

[berkeman]

What have you found so far in your searching? It would seem that if you limit the LHS to integers, there are only solutions for specific choices of c, no?

 

[jedishrfu]

The simplest approach is to find the prime factors of c.

 

[hilbert2]

The easiest way is to just set $a=c$ and $n=1$.

 

[al4n]

The simplest approach is to find the prime factors of c.

thank you. this is very helpful. so I could write something like
aⁿ = 1000
= 2³5³
then like
a = 2^3/n5^3/n
and find values of n that result in whole number exponents. in this case 3, 1.

 

[jedishrfu]

yes or you could look at $a^n = 2^3 * 5^3 = (2 * 5)^3$ and conclude a=10 and n=3

and of course the trivial case of a = 1000 and n=1

 

[hilbert2]

This equation also has the property that there is an infinite number of solutions only when $c=0$ or $c=\pm 1$ (can you prove this formally with mathematical induction or by some other way?).