Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp, prove that either ±a*+1 or ±22k+'a*+l are the square roots of a (mod p).

Q: The number 73 is prime. Enter an a such that 0 <a < 73 and a = 6 (mod 73), if such an a exists.…

A:

Q: 10. Prove that if o is the m – cycle (a,a2 ... am), then for all i e {1,2, ...m}, o'(ar) = ar+i,…

A: Given that σ is the m- cycle (a1,a2,...am). We need to prove (i) σi(ak)=ak+i mod m for all i ∈{1, 2,…

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       =&gt; there exists some integer k such that                    a = b +…

Q: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a E Z+ with a <…

A: We use the properties and definitions of Jacobi symbols to solve this problem. The detailed solution…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: For congruences of primes p = 1 (mod 4), a. If a is a quadratic residue modulo p, and p = 1 (mod 4),…

A:

Q: |Let m be a positive integer. Let a and b be integers such that a = b (modm). Use mathematical…

A: Solution

Q: Suppose a is an integer that is relatively prime to 400. Prove that a40 = 1 (mod 400).

A:

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: 13. Compute 41023 (mod 11). a. 14. Prove that for any integer n, n31 = n(mod7).

A:

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A:

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: Let be a prime such that p = 1 (mod 4). Prove that –1F, is a square in Fp. -

A:

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

A: Solution

Q: Suppose g is a primitive root mod p (with p an odd prime) such that g +1 mod p? Prove g must be a…

A:

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

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: Let a be a quadratic residue of the odd prime p. (1) Prove that a is not a primitive root modulo p…

A:

Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

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 be an odd prime number and an integer g a primitive root with respect to mod p. So is the…

A: This is not a true statement. Since If p be an odd prime and g be a primitive root modulo p. Then…

Q: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

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

A:

Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a € Z+ with a <n.…

A: given data, n=pq , where p and q are distinct primes such that p≡q≡3(mod 4). also aϵZ+ with a&lt;n.…

Q: Let n = 2p for some prime number p &gt; 2. Show that if a is a positive integer such that gcd(n , a)…

A:

Q: show that 2 is a primitive root mod 3^e for all e>=1

A:

Q: 9. Let p be an odd prime, and let ( - 1)(a – 2).. (a - (p - 1)) = "--g12-2 -3 -...+ ap-32 - p-2a +…

A: Let a be any solution of x^2 + 1 ≡ 0 (mod p), so that a^2 ≡ −1 (mod p). Becausep doesn't divide a.…

Q: 1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

A: To prove:                   a265≡amod 105 for all a∈ℤ by using Euler's theorem.   Euler's theorem:…

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Let a be a quadratic residue of the odd prime p. (a) Show that a is not a primitive root modulo p.…

A: An integer a is referred to as quadratic residue modulo p if there exists n∈ℤ such that n2≡amod p.…

Q: 1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let p be an odd prime and r a primitive root of p. Show that if a and b are positive integers in Zp,…

A:

Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

Q: Show that if p is an odd prime, then −1 is a quadratic residue of p if p ≡ 1 (mod 4), and −1 is not…

A:

Q: Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a…

A: According to the given information, Let p is an odd prime and a is an integer not divisible by p…

Q: 4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

A:

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: Use Theorem 1 to determine the number of integral solutions of the polynomial equation a^2 + b^2 =…

A: First of all, find the prime factorization of 23400 as:23400 = 23×32×52×13Since, here

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

A:

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: . Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, ø(m)) = 1, %3D where…

A: Here GCD(b,m)=1 means they are coprime.

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

A:

Q: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: Suppose ab≡0 mod n and gcd(a, n) = 1. Apply Euclid’s Lemma to prove b≡0 mod n.

A: Given: ab≡0mod n and gcd(a,n)=1

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? 1…

A:

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…