Let p be an odd prime. Prove that if g is a primitive root modulo p, then g^(p-1)/2 ≡ -1 (mod p).
2.5. Let p be an odd prime and let g be a primitive root modulo has a square root modulo p if and only if its discrete logarithm log,(a) mod p. Prove t that is even.
Let g be a primitive root modulo to the odd prime p. Prove that: 2)=-1 2)=-1
8. Let g be a primitive root of an odd prime p, and suppose that p3 (mod 4). Show that -g is not a primitive root of p. 8. Let g be a primitive root of an odd prime p, and suppose that p3 (mod 4). Show that -g is not a primitive root of 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.
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,...
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)
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
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....
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration 3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration