Let n ∈ Zp
n+1 ≡ 0 mod p
n≡-1 mod p
n≡ p-1 mod p
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an easy way to find the inverse of p-1 mod p and prove that your method works
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...
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
1. (Complex Multiplication) Let E : y x3 y23 to this congruence mod p. So for example, #E(Z3) = 3 because we have the solutions (0, 0), (1,0) and (2,0) and no more. - x. Then we can reduce E mod p to get mod p for various primes p. We write #E(Z») for the number of solutions This particular equation has some miraculous explore here patterns we (a) Make a chart that lists p, #E(Zp), and #E(Z) - 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....
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3. Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3.
1. Show that a 1728 = 1 (mod p) when p= 7, 13, 19 for all a E N such that p /a. 2. Let p be a prime and p = 3 (mod 4). Show that r2 = -1 (mod p) has no solution. (Hint: Raise both side to (p-1)/2.)
Prove that that if p is a prime such P = 1 (mod 4), then (972) != -1 (mod P).
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,...
Let p be a prime. Show that Zp(X)/(X2+1) is a field iff the equation x2=-1 has no solution (mod p).
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0