Let p be a prime with p ≥ 13. Prove that among the integers 2,11 and 22, either all three are quadratic residues modulo p or exactly one is a quadratic residue modulo p.
Let p be a prime with p ≥ 13. Prove that among the integers 2,11 and...
Please prove the 3 theorems, thank you! 7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p in any complete nesidue system modulo p are quadratic residuess modulo p and half are quadratic non-residues modulo p. From clementary school days, we have known that the product of a pos- itive number and a positive number is positive, a positive times a negative is negative, and the product of two negative numbers is positive....
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...
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 -...
(1) The Legendre symbol and Euler's criterion. (1 pt each) Let p be an odd prime and a Z an integer which is not divisible by p. The integer a is called a quadratic residue modulo p if there is b E Z such that a b2 (p), i.e., if a has a square root modulo p. Otherwise a is called a quadratic non-residue. One defines the Legendr symbol as follows: 1 p)=T-i if a is a quadratic residue modulo...
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...
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...
Let p and n be integers. Prove that, if p is prime, then gcd(p, n) = p or gcd(p, n) = 1. . . (i.) Using proof by contrapositive (ii.) Using proof by contradiction
Let g be a primitive root modulo to the odd prime p. Prove that: 2)=-1 2)=-1
Let J be the ring of integers, p a prime number, and () the ideal of J consisting of all multiples of p. Prove (a) J@) is isomorphic to Jp, the ring of integers mod p. (b) Using Theorem 3.5.1 and part (a) of this problem, that J, is a field.