Prove the following: (a) For an odd prime p, the quadratic residues of p are congruent modulo p to the integers · (P=¹) ². 1², 2², 3²,..., (b) If a is a quadratic residue of an odd prime p, then a is not a primitive root of p.

Q: Show tht if x is an integer, then gcd(x+1, x2-x+1) divides 3.

A: Given that x is an integer. To show :(a) gcd (x+1,x2-x+1) divides 3 (b) To determine for which…

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: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: 3. Define primitive root and find all primitive root of following. (b) 19 (c) 15 25

A:

Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

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

A:

Q: Prove each of the following. (A) Prove that if a and b are quadratic residues modulo m, then ab is…

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: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

A:

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: Consider the following proposed proof, which claims to show that every nonnegative integer power of…

A: Note: we know that in strong induction principle, we assume that the particular statment holds at…

Q: Prove or disprove the following proposition: Let the Jacobi symbol: () 1. %3D Then a is quadratic…

A: a is said to be a quadratic residue modulo n if x2=a(mod n) is solvable for some x.  Otherwise, it…

Q: Prove by Contradiction: There are no integers x and y such that = 4y+2.

A: We have to prove the statement "There are no integers x and y such that x2=4y+2" by contradiction.…

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

A:

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 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: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

A:

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.…