A Wilson prime is a prime, p such that (p - 1)! = -1 (mod p2) For each of the following, determine if it is a Wilson prime: 2, 3, 5, 7, 11

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: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: Use trial and error to find all replacements for x that are less than the modulus that make the…

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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: 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: 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: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: Find the least residue of 448 modulo 7

A: Given problem is : Find the least residue of 448 modulo 7

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: Perform the following congruence computations. Make sure that the number you enter is 20 and ≤N-1,…

A: (a) 5946+7925≡        mod 45⇒13871≡        mod 454530813871-13500371-360011∴5946+7925≡  11      mod…

Q: Let p be prime and a an integer not divisible by p. Prove that if a²"-1 (mod p), then a has order…

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Q: If m>4 is a composite integer, then (m-1)!=1 mod m (m-1)!=2 mod m (m-1)!=0 mod m None of these

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Q: Characterize the primes p mod 20 such ial

A: We have to characterize p mod 20 such that -5p=1.

Q: Perform the following congruence computations. Make sure that the number you enter is ≥ 0 and ≤ N-1,…

A:

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: Let P be an odd prime. Then for all nЄN, we have () = n²¹ (mod p). Р

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Q: (a pour) = (d(1) a) +...+ d& T dG T dl) +04+ mos ad ppo desta sedang

A: Here we use that (p-i)^p=-(i^p) in mod p. We proof this by binomial expansion.

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

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Q: Suppose p is an odd prime. Show that (1² +2 +3P +...+ (p − 1)²) = 0 (mod p).

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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.