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)