please prove lemma and theorems. 8.17 is not needed, thank you
identity 8.17 proved in theorem 8.18. for any doubt write in comment box.
please prove lemma and theorems. 8.17 is not needed, thank you 8.15 Lemma. Let p be...
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...
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...
8.20 Question. Which natural mumbers can be written as the sum of two squares of natural raumbers? State and prove the mast general theorem possible about which natural numbers can be written as the sum of two suares of nutural numbers, and prove it. We give the most gencral result next. 8.21 Theorem. A natural number n can be written as a sum of two squares of natural mumbers if and only if every prime congruent to 3 modulo 4...
please prove the theorems, thank you very much 8.21 Theorem. A natural numbern can be written as a sum of two squares of natural numbers if and only if every prime congruent to 3 modulo 4 in the unique prime factorization of n occurs to an even power Pythagorean triples revisited We are now in a position to describe the possible values for the hypotenuse in a primitive Pythagorean triple. 8.22 Theorem. If (a, h, e) is a primitive Pythagorean...
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 the theorems, thank you 6.1 Theorem. Let anx+an-1- +ag he a polynomial of degree n0 with integer coefficients and assume an0. Then an integer r is a Poot of (x) if and only if there exists a polynomlal g(x) of degree n - with integer coeficients such that f(x) (x)g(x). This next theorem is very similar to the one above, but in this case (xr)g(x) is not quite equal to f(x), but is the same except for the...
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....
please do 7.19 7.20 and 7.21 7.19 Theorem (Quadratic Reciprocity Theorem and q be odd primes, then Reciprocity Part). Let p (e)99 (mod 4) if p (mod 4) or q1 i p 3 (mod 4). (i)) (llint: Iry to use the techniquets used in the case of Putting together all our insights, the Law of Quadratic Reciprocity. we can write one theorem that we call Theorem (Iaw of Quadratic Reciprocity). Let p and q be odd primes, then if p...
please complete exercises 10.4, 10.5, 10.6, 10.7 and 10.9, thank you so much! (I dont understand your comment what is qs 3.6?) 10.4 Exercise. Show that the algorithm descrihed in Question 3.6 for com puting a (mod n) is a polynomial time algorithm in the number of digits in r In the next scrics of problems you will cxplore the usc of this opcration as a means of testing for primality by starting with a familiar theorem. Theorem (Fermat's Little...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...