I'm trying to prove the sum of reciprocals from 1 to (p-1) is divisible by p^2.
Someone showed me how to prove it. But there's a step which I don't grasp. (Or maybe I misunderstood.)
He changed the form of the sum from 1+1/2+...+1/(p-1) to p{1/(p-1)+1/2(p-2)+...+1/{(p-1)/2×(p+1)/2} and first considering only the sum of 1/(p-k)k where k is from 1 to (p-1)/2.
He said in the numerator of the sum of 1/(p-k)k , will appear every term of (p-k)k from 1 to (p-1)/2.
But I cannot show how to gurantee it.
Could you give me any hints or advice?