(b) Prove that n is an odd pseudoprime number if and only if 2"-1-1 mod n.
ly(mod n). 2. Let n > 1 be an odd integer and suppose ? = y2 (mod n) for some x Prove that ged(x - yn) and ged(x + y, n) are nontrivial divisors of n.
Question 1 2(a) Let m>1 be an odd natural number. Prove that 13-5.-(m-2) (- 2-4-6. (-1) (mod m) (m-1) (mod m [Hint : 1 i-(m-1 ) (mod m), 3 Ξ-(m-3) (mod ") , . .. , m-2 1-2 (mod m)] 14 (b) If p is an odd prime, prove that Hint: Use Part (a), and rearrange the Wilson's Theorem formula in two different ways
8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4. (You will need Wilson's Theorem for one (mod p). Prove: a 2--1 mod p has a solution if and only if p dircction of the proof.) 8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4....
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p) g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
Let p be an odd prime. Prove that if g is a primitive root modulo p, then g^(p-1)/2 ≡ -1 (mod p). Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions). Let p be an odd prime. Prove that if g is a primitive...
Prove the following: a. a = b (mod b) implies b =a (mod n) a = b (mod n) and b = c (mod n) imply a = c(mod n)
Suppose that pı, P2, ..., P, are the only primes congruent to 1 (mod 4). Prove that 4p?p, ... p, + 1 is divisible only by primes congruent to 3 (mod 4). Assuming that all odd prime factors of integers of the form x2 +1 are congruent to 1 (mod 4), use Exercise 6 to prove that there exist infinitely many primes congruent to 1 (mod 4).
Discrete Math: Divisibility (Need Help ASAP, will upvote) 1) Prove that if n is an odd positive integer, then n^2 is congruent to 1 (mod 8)
I have first part of question good. Need to prove unique modulo and do not know where to start. Prove that the congruences x-a mod n and x b mod m admit a simultaneous solution if and only if (n, m) | (a -b). Moreover, if a solution exists, then the solution is unique modulo [m, n). Prove that the congruences x-a mod n and x b mod m admit a simultaneous solution if and only if (n, m) |...
p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1 p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1