Prove or disprove: If n is a positive integer, then [itex]n=p+a^2[/itex] where
- [itex]p[/itex] is prime or [itex]p=1[/itex]
Prime is defined such that if p prime and p divides the product ab, then either p divides a or p divides b. Also, primes are irreducible.
Additionally, the fundamental theorem of arithmetic that defines all integers as a unique product of positive primes may be useful.
The Attempt at a Solution
Originally, I had found 25 as a counterexample that cannot be written as the sum of a prime and a square. Then the problem was clarified to include negative primes, and I'm a bit lost as to where I should start. Namely, I'm not sure if I should be working towards proving or disproving the argument. If anyone's worked through this already and could send me some guidance in the right direction, it would be appreciated.