1. If gcd(m,n) = 1, then φ(m,n) = φ(m)φ(n). Use this to give a proof that φ(n) = n Π(1 – 1/p) p/n 2. Prove that d(n) is odd iff n is a perfect square. 3. Prove that σ(n) ≡ d(m)(mod 2) where m is the largest odd factor of n. 3.(2nd Part) If σ(n) = 2n, n is a perfect number. Prove that if n is a perfect number , then ∑1/d = 2. d/n 4.Evaluate σ(210), φ(100) and σ(999). 5.Evaluate d(47), d(63) and d(150).

Arithmetic Functions Combinatorial Study of φ(n) 1. If gcd(m,n) = 1, then φ(m,n) = φ(m)φ(n). Use this to give a proof that φ(n) = n Π(1 – 1/p ...continues

primitive roots

6. Let g be a primitive root of m. An index of a number a to the base... Please see attached.

The Mobius Inversion Formula 1) Prove that ∑ μ(d) φ(d) = Π (2 – p) d/n p/n Primitive roots modulo p 2) Find all primitive roots modulo 5, modulo 9, modulo 11, modulo 13 and modulo 15. Prime Numbers 3) The Fermat numbers are numbers of the form 2^(2^n) + 1 = φn . Prove that if n < m, then φn| φm – 2 4) Prove that if n ≠ m, then gcd (φn, φm) = 1 5) Use the above exercise to give a proof that there exist infinitely many primes.

Theory of Numbers Mobius Theorem ...continues

prime numbers

6. Modify the proof to theorem... to prove that there exists infinitely many prime numbers congruent to 3. Please see attached for full question.

quadratic residues

1. Deduce from the above theorem that if x is sufficiently large, there exists a prime between x and 125x. Please see attached.

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 S ...continues

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.