If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: 3. x* + 1 is reducible over Z, for every prime p. a) True b) False

A: We have to find out whether x4+1 is reducible over Zp for every prime p or not a polynomial is said…

Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

A:

Q: Let a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

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.…

A:

Q: Find the canonical prime factorization of the following using Fermat’s Method and solve for the…

A:

Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: Let p25 be a prime. Prove that p(p+1) (p-1)! +p+1.

A:

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A:

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

Q: Suppose that p is a prime and p ≠ 2. Let a be a nonsquare inGF(p)—that is, a does not have the form…

A:

Q: Show that if p is an odd prime, then n = 2º – 1 passes Miller's test base 2 -

A: Given that n=2p-1 where, pis odd prime. Since, 2p is an even integer. Then, n=2p-1 is an odd…

Q: Find all the positive prime p such that 9p+1 is a perfect cube. Namely, such that there exists an…

A:

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

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

Q: For any distinct primes p, q,r and s, we have pq # rs. For any positive integer n, there exists n…

A:

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A: We have to state whether the statements are true or false

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

A:

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

A:

Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: Let p be a prime number. Find all solutions of the equation xP = x in Zp.

A: We have, xp = x  For x = 0 the above equation will be always true. Hence x = 0 is the answer.

Q: Prove that if a > 1, then there is a prime p so that a < p< a! + 1.

A:

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: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

A:

Q: Find the canonical prime factorization of the following using Fermat's Method and solve for the…

A:

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

A:

Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N).

A:

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: 28. Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(Vd) where d is not…

A: Given [E:D]=2.

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

A:

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: 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: Show that if p is an odd prime, then −1 is a quadratic residue of p if p ≡ 1 (mod 4), and −1 is not…

A:

Q: Suppose that [E:Q] = 2. Show that there is an integer d such thatE = Q(√d) where d is not divisible…

A:

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

A:

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

A:

Q: For every prime p, show that Х-1 — 1 %3 (х — 1)(х — 2) : .. [x — (р — 1)] in Zp[x].

A: Fermat's Little Theorem: Let p be a prime and let a∈N , 0&lt;a&lt;p. Let Ap={1a,2a,....(p-1)a} No…

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

A:

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

A:

Q: Determine whether each of the following is true or false. Prove each true statement, and give a…

A: Read the statement very carefully. Now use the properties and definition to check whether the…

Q: If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

A: Given: G=p.q where p &amp; q are primes To prove: ZG=1 or pq

Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 +3 has roots in Z/61Z by (a): they…

A: Given that, 61 ≡ 1 mod (12) The polynomial x2 + 3 has roots in Z/61Z. The roots are 27 + 61Z and -27…

Q: For every prime p, show that XP-1 – 1 = (x – 1)(x – 2) · · · [x – (p – 1)] in Z„[x].

A:

Q: Prove that if D is a PID and p ∈ D is irreducible, then p is a prime of D

A: