Prove that the only prime of the form n² - 4 is 5.

Q: For which odd primes p is 3 a square mod p?

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Q: A prime of the form 2n − 1 is called a Mersenne prime. (e) More generally, prove that if n is a…

A: Suppose  n  is a composite number,           Say,  n = a.b   for some a, b > 1Claim : cannot be…

Q: There are infinitely many primes of the form 6n+5.

A:

Q: Prove that if p and q are distinct primes, then pq | (p9-1+qp-1-1).

A: p and q are distinct prime. We have to show that pq|pq-1+qp-1-1

Q: Prove that if a and b are two distinct positive primes, then the number k = ab cannot be abundant.

A: To prove that if ( a ) and ( b ) are two distinct positive primes, then the number ( k = ab ) cannot…

Q: For which odd primes p is 5 a square mod p?

A:

Q: 5. Using congruence theory (not brute force), find all solutions to the following linear congruence:…

A: Note:- A two variable linear congruence ax+by≡m    mod (n) has solution if and only if  gcd ( a, b,…

Q: > 2 is a prime, then (12) Prove that if p and (1 2... p-1 p) generate Sp.

A: Solution:                  Let H=1,2,  1,2,3.....n  and H be the smallest subgroupcontaing them.Then…

Q: (ii) The first few cubes are 1,8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, and 1728. Write 1729…

A: The number 1729 is known as a Carmichael number. It can be represented as the sum of two cubes. For…

Q: (c) If p is a prime, then p¹3 will always have exactly fourteen positive divi- sors, namely: {1,p,…

A: If p is a prime, then p13 will always have exactly 14 positive divisors,…

Q: Let n (arak-1 ·*a1a0)3 be the base 3 expansion of a positive integer n. Find the base 3 expansion of…

A: Solution: We know that if for any positive integer r have the base 3 expansion (ap,ap-1,.....,a0)3…

Q: 11. Find a formula for y(n) in terms of the canonical prime factorization of n. Outline at least two…

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Q: Read the copy of the article titled "A New Proof of Euclid's Theorem" by Filip Saidak, which has…

A: As per the question, we are tasked with examining the article "A New Proof of Euclid's Theorem" by…

Q: Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

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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: 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: 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…