Let n be a positive integer with n not equal to 1. Prove that, if n^2+1 is prime number than n^2+1 is expressible in the form 4k + 1 with k an integer.
We can write $\displaystyle n$ in one of the forms: $\displaystyle 4k,4k+1,4k+2,4k+3$. The forms $\displaystyle 4k+1,4k+3$ do not give prime numbers when we compute $\displaystyle n^2+1$. Thus, $\displaystyle n=4k,4k+2$. In both these cases $\displaystyle n^2+1$ has form $\displaystyle 4k+1$.