3. Show that an integer prime p > 0 is expressible as x² + xy + 5y2 if and only if p is 19 or p is a quadratic residue mod 19. (You may use, without proof, that unique factorisation holds in Z[¹+√=1⁹].) 2

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Show that log, q is irrational for any distinct prime numbers p and q. (Distinct here p# q.) means

A: To Determine :-  logp q is irrational for any distinct prime numbers p and q.

Q: Compute 9-¹1 (mod 19) using Fermat's Little Theorem. Show your work.

A:

Q: Work the modular arithmetic problem. (9.7)(mod 5). (9.7)(mod 5) = (mod 5)

A:

Q: Consider the sentence (3xP(x) ^ 3xQ(x)) 3x(P(x) ^ Q(x)). Which of the following is true? A. The…

A: Let P(x) : x is such that x ≡ 1 (mod 3)         Q(x) : x is such that x ≡ 2 (mod 6)    Then,…

Q: There exists an integer m > 3 such that m2 – 1 is prime. The above statement is due to proof by…

A: We have to complete the given sentence.

Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

A:

Q: Can you please show me step by step for this math that makes me undertandable. Q: Determine which…

A: (a) For the congruence 4x ≡ 5 (mod 6), we need to find a solution for x such that when 4x is…

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

A:

Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A:

Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

A:

Q: Work the modular arithmetic problem. (11.10)(mod 7). (11 10) (mod 7) = (mod 7)

A: (a)   (11.10)mod 7  =  110 mod 7                            The remainder when 110 is…

Q: (c) Does the polynomial x² - 27 = 0 (mod 1271) have any roots? Note that 1271 = 31 - * 41 is not a…

A: We use the Chinese Remainder Theorem to find the solutions to the congruence x2-27≡0(mod1271). Now…

Q: Select the following statements that are true. 261000 = 1 (mod 3) (2022) 10 (FC6) 16 In Z6, 2¹⁰ = 2…

A:

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: Prove that for any natural number n, 2(√√n +1 − 1) <1+ 1 √2 + <2√n.

A:

Q: Let a and b be relatively prime positive integers. Show that a(b) + b(a) = 1(mod ab).

A: Solution:

Q: Show that n := e2™i/n is a primitive nth root of unity in C. Verify that (S3 - )? = -3 in C. Verify…

A:

Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: 14. Give a geometric explanation of why 2π/n 1-2 0 r² do = лr² 2 for any positive real number r and…

A: n2∫02πnr2dθ=πr2 For any positive real number r and any positive integer n. Prove this geometrically.…

Q: (a) Prove that 318 = q (mod 19).

A:

Q: In 1732 Euler wrote: "I derived [certain] results from the elegant theorem, of whose truth I am…

A: Details are provided in step 2.

Q: Suppose that p is a prime number and g is primitive root. Prove that the least residues of 4 9²,9ª,…

A: Step 1:To prove that the least residues of g2,g4,g6,gk,...,gn−1 are all quadratic residues modulo n,…

Q: 3. Prove that if m is a positive integer and x is a real number, then 1 2 m 1 [mx] = [x] + \x + = |…

A: The greatest integer function is one of the most important real valued number theoretic function.…

Q: Work the modular arithmetic problem. (8.9)(mod 4). (8.9)(mod 4) = (mod 4)

A:

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Use the factorization A = PDP' to compute A", where k represents an arbitrary positive integer. -2…

A:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: 6. If |G| = pq for distinct primes p and q, show that G is not simple. (Note the slight difference…

A: