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

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

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Q: 7) Let p be an odd prime, find if a) p =1 mod 8

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Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

A:

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: Determine the smallest integer k > 1 such that ak = 1 modulo 2465 for all in- tegers a which are…

A: A composite number m is called Carmichael number if for all integers b coprime with m, bm-1≡1modm…

Q: Let p be an odd prime number and an integer ga primitive root with respect to mod p. So is the…

A: We will use basic knowledge of elementary number theory to solve this question.

Q: If p is a prime with p=3 mod 4 and a is quadratic residue mod p, then -a is a quadratic residue mod…

A:

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

A:

Q: Let n be a positive integer, and a an integer Prove that if the order of a modulon is n-1 then n is…

A: The given problem is to prove that n is prime from the given problem given a modulo n is set of…

Q: 8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1…

A: a) Given that p and q are odd primes and qap-1. So, ap≡1mod q If p=1 then  a≡1mod q So, qa-1 If p≠1…

Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p =&gt; αp-1…

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: Let n be a positive integer, and a an integer Prove that if the order of a modulo n is n-1 then n is…

A: Problem 2.  Solving it using the concept of phi function

Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

Q: Let n be an integer greater than 1. If (n-2)! ≡ 1 mod n, then Prove n is a prime number.

A: Use the Wilson theorem and different case to solve this problem as in next step.

Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

A:

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: Suppose r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =

A: Given: p≡3mod 4 To find: ordpp-g=p-12

Q: Let p be an odd prime, and let g be a primitive root modulo p. Prove that -g is a primitive root…

A: Given : p= odd prime g= primitive root modulo p

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: (g) There are exactly six prime numbers p with the property that x81 = x (mod p) for every integer…

A: Given that, x81≡xmod p Find the six prime which satisfies the given congruence. Fermat's Theorem: If…

Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

A:

Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

A:

Q: Let p be an odd prime number. Prove that [(3)=1] → [p = 1 mod 6).

A: Introduction: An integer-based arithmetic system that takes the remainder into account is called…

Q: Let odd prime number P =4 (mod L), be an ond If the integer g iso primitive root with respect to mod…

A: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). We have to show that −r is also a…

Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: Use Dirichlet's theorem to show that there are infinitely many primes whose dicimal expansion ends…

A:

Q: (a) Suppose p is prime, pła and k = gcd(n, p – 1). If p-1 a k =1 (mod p), prove that the congruence…

A:

Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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

A:

Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: A Wilson prime is a prime, p such that (p - 1)! = -1 (mod p2) For each of the following, determine…

A: Introduction: The only two elements in prime numbers are 1 and the number itself. When we consider…

Q: 2. Let p and q be distinct primes. Let a be an integer such that aº = a aª = a mod p. Prove that aPª…

A: ap = a (mod q) and aq = a (mod p)

Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

A:

Q: For every prime p, show that ( p - 2)! mod p = 1.

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Q: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). Show that −r is also a primitive root…

A: The given problem is to show that -r is the primitive root of p when r is a primitive root of prime…

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

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Q: Find all three incongruent solutions to the congruence x³ = 1 (mod 103). Express these solutions as…

A: The solutions are obtained by knowing the definition of primitive roots and applying them carefully.…

Q: 3. Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Step:-1 Since, we know for a prime p, modulo p forms a cyclic group. And, also according to a known…

Q: Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public…

A: RSA algorithm: 1. Choose two distinct prime numbers p and q. Take n = pq. 2. Compute λ(n), where λ…