By Waclaw Sierpiński
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However, if we carry over the operations in the product (1 − f (p1 ))(1 − f (p2 )) · · · (1 − f (pk )), we get a sum of the form (−1)k f (pq11 )f (pq22 ) · · · f (pqkk ) = qi =0 or 1 (−1)k f (pq11 pq22 · · · pqkk ). qi =0 or 1 Hence, by (2) it is evident that k (−1)k f (pq11 pq22 · · · pqkk ) (1 − f (pi )) = qi =0 or 1 i=1 ✷ μ(d)f (d). 6. 3. 1. The Euler function φ(n) is deﬁned as the number of positive integers which are less than or equal to n and at the same time relatively prime to n. Equivalently, the Euler function φ(n) can be deﬁned by the formula n φ(n) = m=1 1 1 .
Fn | (2Fn −1 − 1). Fn | 2(2Fn −1 − 1) or Fn | (2Fn − 2). 4. Fermat numbers are coprime. Proof. Let us suppose that Fermat numbers are not coprime. Then, there exists a prime number p, for which p | Fm and p | Fn , for some positive integers m, n. Without loss of generality we may suppose that n < m. 2 we obtain Fn | Fm − 2 and thus p | Fm − 2. But, because of the fact that p also divides Fm , we get p | Fm − (Fm − 2) and consequently p = 2. But, that is a contradiction since Fermat numbers are odd integers and therefore are not divisible by 2.
But, generally it is true that if α ≡ β (mod γ1 ) and α ≡ β (mod γ2 ), with gcd(γ1 , γ2 ) = 1, then α ≡ β (mod γ1 γ2 ). Therefore, since gcd(pk11 , pk22 , . . , pknn ) = 1, we obtain aφ(m) ≡ 1 (mod m). ✷ The above theorem is a generalization of Fermat’s Little Theorem and was ﬁrst proved by Leonhard Euler in 1758. 4. Let a, b, c ∈ Z, where at least one of a, b is nonzero. If d = gcd(a, b) and d | c, then the diophantine equation ax + by = c has inﬁnitely many solutions of the form b x = x0 + n, d a y = y0 − n, d where n is a positive integer and (x0 , y0 ) is a solution of the equation.