Thread: Finding the biggest possible value of n

Let n satisfy the equation, a^n+b^n=c^2. When a,b,c are prime numbers and n is an integer, there is one solution at least for the equation.

How to find the biggest possible value of n?

Could you give us the solution that you say exists?

Have you managed to narrow down the possibilities for a and b?

I have made some progress but have not found a solution.

I think i solved it



And take Mod 5 in the last equation

Originally Posted by Mhmh96

I think i solved it

What's your solution?

Originally Posted by Mhmh96

And take Mod 5 in the last equation

Why?

How about 2+2=4?

It's really easy to show that one of a,b must be 2.

Then use casework on the parity of n. Note that if n is odd, the LHS factors.