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: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

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Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

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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: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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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: Let p be an odd prime. Prove that (a) If p = 1 (mod 5) or p = 4 (mod 5), then 5 is quadratic residue…

A: Given p is an odd prime. We have to prove (a). if p≡1(mod 5)  or  p≡4(mod 5), then 5 is a…

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: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

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…

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Q: 4. Let p be an odd prime. (a) For any integer 1 <k <p-2, show that (+1) = (#¹) where k denotes the…

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Q: Prove Euler’s criterion, which states that if p is an odd prime and a is a positive integer not…

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Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: a)Consider the following data: 3 is a quadratic residue for 23 and 37. 3 is a quadratic nonresidue…

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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: Let p and q be two distinct primes. (a) Prove that if a = b (mod p) and a = b (mod q), then a = b…

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Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: (c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n

A: We know that,

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

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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: There are Infinitely many Primes.

A: To prove that there are infinitely many Primes.

Q: 2. Let p be a prime. a) Prove that if p > 3, then p is of the form 6k+ 1 or 6k + 5. b) Prove that if…

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Q: Let me Z+ be odd with prime-power factorization m = p₁¹p2²p3³p4, where a1a2 a3 a4 P₁ = P2 = P3 = ±5…

A: As per the question we are given a odd numbers m and its prime power factorization as m =…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

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&gt;5 and see which prime works.

Q: Let c be an element of N, and p be a prime. Prove that if c is greater than or equal to 3, then c^p…

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

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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: 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: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

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Q: Starting with (19), generate an "infinite" number of primes using Euclid's method for primes of the…

A: We imitate Euclid's method and solve the problem.

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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