Problem 6: Let p be an odd prime number, so that p= 2k +1 for some...
Let p be an odd prime. Write p in the form p = 2k + 1 for some k E N. Prove that kl-(-1)* mod p. Hint: Each j e Z satisfies j (p-od p.
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...
Problem 7. Let M = 2" – 1, where n is an odd prime. Let p be any prime factor of M. Prove that p=n·2j + 1 for some positive integer j.
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 an odd prime and a an integer with p not dividing a. Show that a(p-1)/2 is congruent to 1 mod p if and only if a is a square modulo p and -1 otherwise. (hint: think generators)
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...
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
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.