Let p be a prime number. Prove that 19–1 + 2P-1 + ....(p – 1)p-1 =...
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...
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,...
4. (a) [3] Let p be prime and let M, denote the number 2P – 1. The number M, is called a Mersenne number, and if it is prime, it is called a Mersenne prime. There is a test, called the Lucas-Lehmer Test, that gives a necessary and sufficient condition for My to be prime. It is always used to verify that a Mersenne number, suspected of being prime, is indeed a Mersenne prime. Give the statement of this test....
ame: . (10 points) Let p > 3 be any prime number. (a) Show that p mod 6 is equal to 1 or 5 (b) Use part (a) to prove that pe - 1 is always a multiple of 24.
Write your own answers 10. Let p be a prime number, and let a be an integer that is not divisible by p. Prove that the congruence equation ax = 1 mod p has a solution X e Z.
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.
Let p be a prime number. Prove that if there exists a solution to the congruence c(mod p), then there exist integers m, n such that p = m2 + 2n2. Hint. Make a careful study of the proof of Fermat's Two Square Theorem, and then try to modify that proof (or at least some portion of it) to come up with a proof of this statement. Toward the end of the proof, the following observation can be helpful: If...
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 -...
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...
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....