If p and q are odd primes, then a)-4 is a primitive root of q. b) 4 is a primitive root of q. c) (p-1)/4 is a quadratic residue of q. d) None of the above.

Q: 2. Exercise §4.3 #36c. Find congruences characterizing all prime numbers p for which the following…

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Q: Prove that pqr is a deficient number, where p, q, r are three distinct primes with p <q < r

A: We have to prove that pqr is deficient number where p, q and r dustinct primes  Such that p &lt;  q…

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contradiction.

A: Given: 3n + 2 is an even integer Let n is not even, thus n is odd. Therefore, by using the…

Q: Prove that there do not exist three consecutive integer values of nn for which 43n+1443n+14 is a…

A: Given 43n+14 We have to Prove that  ∄  three consecutive integer values of n for which 43n+14 is a…

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) that 250 758…

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Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: Characterize the primes p for which 63 is a quadratic residue.

A: Quadractic residue: Let p  be any odd prime number and gcd(a,p)=1. Then a is said to be a quadratic…

Q: The general combinatorial formula for the nth triangular number (1, 3, 6, 10, 15, 21, .) is On+1Cn…

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Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) The 758 ppose

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Q: primes

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Q: 5 is a quadratic non- residue mod 17 True False

A: 5 is a quadratic non- residue mod 17

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: 9. Prove that V2 is an irrational number.

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Q: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: A. What are quadratic residues and nonresidues ? B. Write down the definition of elite primes in…

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Q: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

A: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

Q: 1. Give the ten 10th roots of unity. Write them in the form e. Among these 1Oth roots, which are the…

A: According to Bartleby's guidelines, we are allowed to answer only one question at a time. So here is…

Q: By Principle of Mathematical Induction prove 6n 6

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Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: 1. Prove that V3 is irrational number.

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Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: (a) Characterize the primes p for which 63 is a quadratic residue.

A: Quadratic residue: Let p be any odd prime member and gcd(a, p)=1. Then a is said to be a quadratic…

Q: #3. Let n be an integer with nz1 and Let b, bz . bn be ... positive real numbers, not all egual,…

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Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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Q: 2. Find all Pythagorcan Triples (not just primitive ones), containing the integer 15.

A: The Pythagorean triples are positive integers a,b,c such that a2+b2=c2. We can say that the…

Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: Suppose that p and q = 4p+ 25 are both prime. Show that p is a quadratic residue mod q.

A: Given p and q are both prime. and q=4p+25 To show that p is a quadratic residue mod q.

Q: Do you think it is worthwhile, or useful, to solve the puzzles proposed in Euclid's theorem of…

A: The objective is to solve the puzzles proposed in Euclid's theorem of primes There are infinitely…

Q: Prove that for any odd prime p different from 3, ) = 1 if and only if p = ±1 (mod 12). (hint: Use…

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Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

Q: Prove in a thoroughly written proof that for all integers a  and m, if a and m are the lengths of…

A: Recall the following. Pythagoras theorem: For any right triangle, the square of the hypotenuse is…

Q: Show that if p is an odd prime, then there are exactly (p − 1)∕2 quadratic residues of p among the…

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Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

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Q: ) Prove that for all positive integers k, 22k – 1 is a multiple of 3.

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

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: prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

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Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

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Q: Let p and q be odd primes. Prove that (p^2) q is not a perfect number. [Hint: Factor 1 + p + q + pq…

A: Let σp2q=σp2σq Then we have  σp2q=σp2σq=1+p+p21+q=p3-1p-1q2-1q-1≤p3p-1q2q-1

Q: infinitely many primes of the form

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Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Let p and q be primes. Prove that pq +1 is a square if and only if p and q are twin primes.

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Q: Prove for every pair p, q of distinct primes that

A: To prove that for every pair p, q of distinct primes that pq is irrational.

Q: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

Q: 1. Prove that 3 is irrational number.

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Q: True or False 19 (a) By the law of quadratic reciprocity, (3) = (²9). 19 (b) If a is a quadratic…

A: We will use the basic knowledge of quadratic residue and Legendre's symbol to answer this question…

Q: 3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n+ 6. Hint: Recall…

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Q: 3. Prove that if n is a triangular number, then so are 9n +1, 25n + 3, and 49n + 6. Hint: Recall…

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