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)
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....
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...
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).
Need help!! Please help — crypto math 1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd? 1. Determine L13(18) for p 19. 2. Let p be prime,...
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai? 5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
Let p be a prime number. Prove that 19–1 + 2P-1 + ....(p – 1)p-1 = -1 mod p
Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting sequence is periodic with smallest period p(p - 1). (This means that p(p - 1) is the least among all positive integers l with the property that whenever n = m (mod l), we have n" = m" (mod p).) Let p be a prime. Consider the sequence 11,22,3, 44,55 modulo p. Prove that the resulting sequence is periodic with smallest period p(p -...
76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2 = -1 mod p. 77.let p be an odd prime. Prove that if Ord, (a) = 3, then 1+ a + a? = 0 mod p and Ord,(1 + a) = 6. 78.Show that 3 is a primitive root modulo 17. How many primitive roots does 17 have? Find them.
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