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MathYou can use modular arithmetic to show this. Firstly, 105 = 3 x 5 x 7 using prime factorisation. We can then write that `2^105 + 3^105 = ((2^3)^5)^7 + ((3^3)^5)^7` 1) When taking modulus 5 of this...

MathEuler's product formula states that `prod_(p quad "prime") 1/(1p^(s)) = sum_(k=1)^infty 1/k^s` So if we have that `sum_(k=1)^infty 1/k^2 = prod_(p quad "prime")1/(1p^(s))` for `s in...

MathEuler's Theorem says that if `a` and `n` are relatively prime, or coprime, then `a^(phi(n)) = 1` `mod n` where `phi(n)` is Eulers phi function (this gives the number of numbers less than or...

MathThe definition of Euler's `phi` function is that `phi(n)` equals the number of positive integers less than or equal to `n` that are coprime, or relatively prime to `n`. Coprimes have a greatest...

MathEuler's Theorem states that if `a` and `n` are coprime then `a^(phi(n)) = 1` `mod n` ` ` ` `where `phi` is Euler's `phi` function (which gives the number of numbers less than or equal to `n`...