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

Q: Let p be a prime. What is the remainder of (p−1)!, when divided by p? In other words: find a simple…

A: Remainder will be (p-1). As p is a prime number. p is divisible by either p or 1. It will never get…

Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

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: True or false: If p is an odd prime, then (p-1)! + 2^(p-1) congruent 0(mod p). Justify your answer.

A: p is an odd prime.

Q: (Wilson’s Theorem) For every integer n > 1, prove that (n - 1)!mod n = n - 1 if and only if n is…

A: We have to prove: For every integer n>1,n-1! mod n=n-1 if and only if n is prime.

Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

A:

Q: Prove that If a is an element of order n and p is prime to n, then a\power{p} is also of order n.

A: is prime to We have to show

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: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

A:

Q: Let p be an odd prime, let a be an integer that is not divisible by p, and let b be a square root of…

A: We present the ideas involved in the problem. Basic Definition of quadratic residues and all will be…

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: 2. Find 52019 (mod 13).

A: Here,given  52019(mod 13)Now we find 52019(mod 13) using Chinese Remainder Theorem and Fermat's…

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: 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: (8) Using that 151 is a prime (so that, in particular, Fermat's Little The- orem can be used; hint,…

A: That's easy. Have a great day!!!

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>5 and see which prime works.

Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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: 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: