let the symbol = denote congruence (a=b mod n means a is congruent to b mod n)

let p be a prime number, m,k be two positive integers, p does not divide a

Prove that if p>2 or m>1 and $\displaystyle a=1 mod p^m$ is true, a=1 mod p^(m+1) is false, then:

a^(p^k)=1 mod p^(m+k) is true (I have already proved this)

a^(p^k)=1 mod p^(m+k+1) is false (this is the one I need help on)

thank you! and sorry about the sloppy notation, I don't know how to do the imaging stuff