By Atle Selberg
Atle Selberg is popular for his profound paintings in quantity conception and different branches of arithmetic, highlighted, probably, by way of his robust hint formulation. he's additionally identified for the $64000 lectures and papers that he hasn't ever released. With quantity II of Selberg's gathered Papers, those unpublished works develop into extensively on hand for the 1st time. This assortment will fill a long-standing hole within the literature.
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Extra info for Atle Selberg Collected Papers, Volume 2
N. 46 4 Congruences The integers xi , x0 are distinct solutions and thus xi ≡ x0 (mod p). Therefore, p | p(xi ), i = 1, 2, . . , n, and consequently the polynomial congruence p(x) ≡ 0 (mod p) has n solutions, which is impossible since the polynomial p(x) is of n−1 degree and we have assumed that the theorem holds true for polynomials of degree up to n − 1. Hence, by the Mathematical Induction Principle, it follows that the polynomial congruence an xn + an−1 xn−1 + · · · + a1 x + a0 ≡ 0 (mod p), where an ≡ 0 (mod p), has at most n solutions.
2n−1 p are the desired divisors. Hence, we get σ1 (2n−1 p) = 1 + 2 + 22 + · · · + 2n−1 + p + 2p + 22 p + · · · + 2n−1 p = (p + 1)(1 + 2 + 22 + · · · + 2n−1 ) = (p + 1)(2n − 1) = 2n (2n − 1). Therefore, σ1 (2n−1 p) = 2n (2n − 1), ✷ which is the desired result. 3 (Euler). Every even perfect number can be represented in the form 2n−1 (2n − 1), where n is a positive integer and 2n − 1 is a prime number. Proof. Let k be a perfect number and n − 1 be the greatest power of 2 which divides k. Then, for some positive integer m it holds 2k = σ1 (k) = σ1 (2n−1 m) = σ1 (2n−1 )σ1 (m), since 2n−1 and m are relatively prime integers.
Mk , a1 , a2 , . . , ak ∈ Z, such that gcd(mi , mj ) = 1, for i = j and gcd(ai , mi ) = 1, for every i, where 1 ≤ i, j ≤ k. If m = m1 m2 · · · mk , then the system of linear equations a1 x ≡ b1 (mod m1 ) a2 x ≡ b2 (mod m2 ) .. ak x ≡ bk (mod mk ) has a unique solution modulo m. Proof. At ﬁrst, we shall prove that the system of linear congruences has a solution modulo m and afterwards we shall prove the uniqueness of that solution. Set ri = m/mi . Then, it is obvious that gcd(ri , mi ) = 1 and thus, the linear congruence ri x ≡ 1 (mod mi ) has a unique solution.
Atle Selberg Collected Papers, Volume 2 by Atle Selberg