Question 1 2(a) Let m>1 be an odd natural number. Prove that 13-5.-(m-2) (- 2-4-6. (-1)...
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...
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...
Problem 5 1. Find the values of (379) and (4725). 2. Prove that for any m > 2, (m) is even. 3. Prove that if (371) - 36(n) then 3|n. Hint: Try proving the contrapositive. 4. Suppose that a =b (mod m), a = b (mod n), and ged(m, n) = 1. Prove that a = b (mod mn). 5. Use Euler's Theorem and the method of successive squaring to find 56820 (mod 2444). That is, find the canonical residue...
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.
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...
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.
Question 6 (optional) For positive integers p 2 2, Wilson's Theorem states that p is a prime if and only if (p-1)!-1 (mod p) (a) Prove Wilson's Theorem (b) Discuss whether Wilson's Theorem is suitable as a primality test for finding primes to use with RSA. Question 6 (optional) For positive integers p 2 2, Wilson's Theorem states that p is a prime if and only if (p-1)!-1 (mod p) (a) Prove Wilson's Theorem (b) Discuss whether Wilson's Theorem is...
Problem 6: Let p be an odd prime number, so that p= 2k +1 for some positive integer k. Prove that (k!)2 = (-1)k+1 mod p. Hint: Try to see how to group the terms in the product (p − 1)! = (2k)! = 1 * 2 * 3... (2k – 2) * (2k – 1) * (2k) to get two products, each equal to k! modulo p.
Problem 3 1. Find the values of (379) and (4725). 2. Prove that for any m > 2, (m) is even. 3. Prove that if (371) - 36(n) then 3|n. Hint: Try proving the contrapositive. 4. Suppose that a =b (mod m), a = b (mod n), and ged(m, n) = 1. Prove that a = b (mod mn). 5. Use Euler's Theorem and the method of successive squaring to find 56820 (mod 2444). That is, find the canonical residue...