Let p be a prime number. Prove that if there exists a solution to the congruence...
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 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.
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....
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.
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 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 -...
please prove lemma and theorems. 8.17 is not needed, thank you 8.15 Lemma. Let p be a prime and let a be a natural number not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lyl 8.16 Theorem. Let p be a prime such that p (mod 4). Thenp is equal to the sum of two squares of natural numbers. (Hinl: Iry applying the previous lemma to a square root of- mohulo...
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.
8.15 Lemma. Let p be a prime and let a be a natural mumber not divisible by p. Then there exist integers x and y such that ax y (mod p) with 0xl.lylP 8.16 Theorem. Ler p be a prime such that p 1 (mod 4). Then p is equal to the sum of two squares of natural numbers. (int: Iry applying the previous lemma to a square root of- mochdo p.) Knowing which primes can be written as the...
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...