Prove that an odd integer n 1 is prime if and only if it is

Prove that an odd integer n > 1 is prime if and only if it is not expressible as a sum of three or more consecutive positive integers.

Solution

For this , If k were a prime p that could be expressed this way, then you\'d have (nm)(n+m+1)=2p. But nm3, and n+m+1 would only be bigger than that. Since 2p has only the factors 2 and p.

So suppose k is an odd non-prime, which you can write as k1k2 where k1k2 are odd numbers that are at least 3. You now want to solve (nm)(n+m+1)=2k1k2. It\'s natural to set nm=k2 (the smaller factor), and 2k1=n+m+1, the larger factor. Solving for n and m one gets n=2k1+k21/2 and m=2k1k21/2. Since k1 and k2 are odd these are both integers. And since k1k2, the numbers m and n are nonnegative.