If p and q are odd primes, then a) -4 is a primitive root of q. b) 4 is a primitive root of 9. 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: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

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Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

Q: (2) Prove by contradiction that if 3z2- is irrational then r is irrational. 215

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Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

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

A: 5 is a quadratic non- residue mod 17

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: 1. Prove that the Mersenne number M13 is a prime; hence, the integer n = 2' (213 – 1) is perfect.…

A: handwriting solution

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

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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: For p an odd prime in Z , Prove: If p is not prime in Z[i] then -1 is a perfect square in Z/pZ

A: Given p is an odd prime in Z.Assume that (-1) is a perfect square in Z/pZ.Then, there exists an…

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: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

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Q: Prove that a positive integer n is prime number.if no prime p less than or equal to n divides n.

A: Let us assume that the contradiction of the statement , let  n ≥ 0 be a composite number .  Thus , n…

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: Prove the following statement directly from the definitions. The product of any two even integers is…

A: The product of any two even integers is a multiple of 4.

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: Show that square root P is irrational where P is prime by using contradiction.

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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: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

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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: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

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: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

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: Find (with proof) all positive primes p such that 217p is a Carmichael number.

A: Carmichael number - A number n is a Carmichael number if and only if n=p1p2…pj , a product of at…

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

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Q: infinitely many primes of the form

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Q: 1. Use the approach of the Great Theorem of Chapter 10 to determine if 1,679,617 is prime just as…

A: Hello thankyou for the question. According to our honor code can answer only three sub parts at a…

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: Prove that the messenger number M19 is a prime hence the integer n = 218 (219 -1) is perfect

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Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Prove that there are infinitely many primes of the form p = 4n + 3.

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Q: 1. Prove that 3 is irrational number.

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Q: Prove by induction that the sum of the first n odd numbers is n². In other words, prove E1(2i – 1) =…

A: To prove by induction that the sum of the first natural numbers is n^2.

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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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.