Hi,

i need to verify if my proof for this particular problem is correct.

The theorem:

Let m, n be relatively prime positive integers. Prove that m^Φ(n) + n^Φ(m) Ξ 0mod mn

Proof:

nx = m^Φ(n) – 1

my = n^Φ(m) – 1

Let xy = z

nm(z) = (m^Φ(n) – 1)( n^Φ(m) – 1)

nm(z) = m^Φ(n) n^Φ(m) - m^Φ(n) - n^Φ(m) + 1

m^Φ(n) + n^Φ(m) - m^Φ(n) n^Φ(m) – 1 = (nm)(-z)

m^Φ(n) n^Φ(m) Ξ 0mod (mn)

Therefore,

m^Φ(n) + n^Φ(m) Ξ 1mod (mn)