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: 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: 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: 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: Prove Wilson's theorem that, for any prime p, (p — 1)!= -1 (mod p).

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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: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

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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: 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: 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: Let n be a positive integer and p a prime that divides n. Prove that pis prime in Zn.

A: It is given that, n be a positive integer and p be a prime that divides n.  We have to prove that, p…

Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

A: The proof is based on a simple application of the law of quadratic reciprocity and the…

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,…

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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: Find all natural numbers b with 100 < b < 200 such that: 5211 b (mod 35)

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

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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: (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: Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Given that p is prime . Let us take p=19 By Fermat's little theorem, " If p is a prime and a is any…

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: Prove that for any odd prime p and a, b = Z, ap+bP = 0 (mod p) ⇒ ap+bP = 0 (mod p²).

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Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

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

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Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

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Q: If possible, let berational. v2 a Then, there exist positive co -primes a andb such that - -

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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.

Q: Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

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Q: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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