11. Find a formula for y(n) in terms of the canonical prime factorization of n. Outline at least two different proofs that this formula is valid.

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: (6) In your computations below, do the work "without a calculator," i.c., do your computation in a…

A: One may find here that

Q: (c) Prove that m is prime if and only if y(m) = m - 1.

A:

Q: (a) Prove that for every integer n ≥ 2, 1 n+2 1 + n+1 +...+ 2n

A: Since you have posted a multiple question according to guildlines I will solve first (a)question for…

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

A:

Q: Given n > 1, let p be a prime number such that n < p < 2n (by the way, 2n there is always a prime…

A: here we need to expand  by using  nr=n!r!×n-r! and then show that expression obtained have a factor…

Q: 6. Prove that m is prime if and only if y(m) = m - 1.

A:

Q: Give a formula for o(n) as a product over the distinct primes dividing n.

A:

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: (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: (b) There exists an integer b such that for all integers n > b, n³ – 8 is prime.

A:

Q: Provide a definition of a cubic residue of an odd prime p similar to the definition of a quadratic…

A:

Q: Find the smallest positive integer N≥ 2 that gives remainder 1 upon division by each of the integers…

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: 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: 11. Prove metic them t of type

A:

Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: (b) Find a prime p> 5 such that 2 and 3 are both primitive roots modulo P.

A: We test all the primes p&gt;5 and see which prime works.

Q: .. (a) Prove by mathematical induction that 9**2 – 8n – 17 is divisible by 64 for all positive…

A:

Q: For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive…

A: Let's break down the problem into two parts: (a) For every natural number 𝑥, there is a natural…

Q: Let p ∈ Z be a prime integer. (a) Show that if p > 3, then p is of the form 6k + 1 or 6k −1 for some…

A: Answer

Q: Ulia primeive Toute 95. Find all primes p ≤ 29 such that 2 is a primitive root modulo p.

A: Here we have to find all primes less than 29 or equal to 29 having primitive root 2.

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: