quadratic residues 2

6. Use Gauss' Lemma to show that 17 is a quadratic residue module 19. Please see attached.

1. Let N1(p) denote the number of pairs of integers in [1, p – 1] in which the first is a quadratic residue and the second is a quadratic nonresidue modulo p. Prove that N1(p) = (1/4) (p – ( – 1)^((p – 1)/2)) 2. Let N2(p) denote the number of pairs of integers in [1, p – 1] in which the first is a quadratic nonresidue and the second is a quadratic residue modulo p. Prove that N2(p) = (1/4) (p – 2 + ( – 1)^((p – 1)/2)) 3. Let N3(p) denote the number of pairs of integers in [1, p – 1] in which the first is a quadratic nonresidue and the second is a quadratic nonresidue modulo p. Prove that N3(p) = (1/4) (p – 2 + ( – 1)^((p – 1)/2)) 4. Use the results of theorem 10-4 and corollary 10-1 to construct solutions of x^2 + y^2 =29. 5. Prove ( without assuming corollary 10-1) that, if p is a prime ≡ 1 (mod 4), then there exists positive integers m, x, and y such that x^2 + y^2 =mp, with p ┼ x, p ┼ y, 0 < m < p [ Hint: use the proof of Theorem 11-2]. Theorem 10-4: If p is an odd prime, then ν(p) = (1/8)p + Ep where | Ep| < (1/4)(p)^(1/2) +2. Corollary 10-1: Every prime p ≡ 1(mod 4) is representable as a sum of two squares. Theorem 11-2: For each prime p there exist integers A,B and C, not all zero, such that A^2 + B^2 + C^2 ≡ 0 (mod p).

Theory of Numbers The Distribution of Quadratic Residues Sums of Squares Sums of Four Squares 1. Let N1(p) denote the number of pairs of integers in [1, p – 1] in which the first is a quadratic residue and the second is a quadratic nonresidue modulo p. Pr... click for more

Math proofs

Consider the compound statement... a) Find the truth table for the statement. b) IS the statement a tautology? c) In the following... Please see attached.

Number Theory

1) Let e = ... be an RSA enciphering exponent. Prove that, for any... Please see attached.

Superincreasing Sequence; Prove that ... is a Prime

1) Let S= {see attachment} satisfies (see attachment) >2b for all j =1,2,3,…….n-1. Prove that S is a superincreasing sequence. 2)Prove that n ... Please see attachment for complete set of questions. Thanks.