Congruences, Primitive Roots, Indices and Table of Indices

6. Let g be a primitive root of m. An index of a number a to the base (written ing a) is a number + such that g+≡a(mod m). Given that g is a primitive root modulo m, prove the following... 7. Construct a table of indices of all integers from.... 8. Solve the congruence 9x≡11(mod 17) using the table in 7. 9. ...continues

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, Congruences, Prime Factorization and Quadratics

6. Modify the proof to theorem... to prove that there exists infinitely many prime numbers congruent to 3(mod 4) 7. Suppose that p1...pn are the only primes congruent to 1(mod 4), prove that ....is divisible only by primes congruent to 3(mod 4) 8. Assuming that all odd prime factors of integers of the form x^2 + 1 are cong ...continues

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

Bijections, Denumerability, Tautologies, Sets, Logic and Proofs (7 Problems)

Consider the compound statement (P ^ Q) V (~ P ^ R) a) Find the truth table for the statement. b) IS the statement a tautology? c) In the following program code, what has to be the output for the answr to be "yes"? ... 2. Given the following sets A = {1.2,3.4}, B = {2,3,4,5}, and C= {2,4,6} a) Find .... 4. Prove that ...continues

Number Theory : RSA Enciphering Exponent

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.

Prime and Nonprime Numbers : Multiplication

Does a prime number multiplied by a prime number ever result in a prime - Why? Does a nonprime multiplied by a nonprime ever result in a prime - why? Is it possible for an extremely large prime to be expressed as a large integer raised to a very large power? Explain. Are there infinitely many natural numbers that are not pri ...continues