Create and solve two assignments that represent the theorem 11.3 (b) q| M,+2, provided that q 3 (mod 8) or q=5 (mod 8).Mersenne numbers are prime or composite. One such test is presented next.determining whether certain special types iTheorem 11.3. If p and q = 2p +1 are primes, then either q | M, or q | Mp+2, butnot both.Bnof With reference to Fermat's theorem, we know that29-1 -1 = 0 (mod q)and, factoring the left-hand side, that(29-1)/2 – 1)(29-1)/2 + 1) = (2P – 1)(2P + 1)= 0 (mod q)What amounts to the same thing:Mp(M,+2) = 0 (mod q)The stated conclusion now follows directly from Theorem 3.1. We cannot have botha|M, and q | Mp+2, for then q | 2, which is impossible.A single application should suffice to illustrate Theorem 11.3: if p = 23, then0= 2p +1 = 47 is also a prime, so that we may consider the case of M23. Thequestion reduces to one of whether 47 | M23 or, to put it differently, whether 223 =1 (mod 47). Now, we have223 = 2 (2°)ª = 2°(-15)*(mod 47)But(-15)* = (225)² = (-10)² = 6 (mod 47)%3DPutting these two congruences together, we see that223 = 23.6 = 48 = 1 (mod 47)whence M23 is composite.We might point out that Theorem 11.3 is of no help in testing the primality of29, say; in this instance, 59 M29, but instead 59| M29 +2.the two possibilities q| M, or q| Mp+2, is it reasonable to ask: Whatnuitions on q will ensure that a | M,? The answer is to be found in Theorem 11.4.Theorem 11.4. If a = 2n +1 is prime, then we have the following:%3D91M provided that q1 (mod 8) or q 7 (mod 8).Proof, Toaouivalent to asserting that

Answered: Image /qna-images/answer/f5991fc1-6c11-48f6-a2e2-0175fefdacb7.jpg

Theorem 11.3. If p and q = 2p+1 are primes, then either q | Mp or q |Mp +2, but presented next. not both. Proof. With reference to Fermat's theorem, we know that 29-1 -1 = 0 (mod q) and, factoring the left-hand side, that (29-1)/2 - 1)(29-1)/2 + 1) = (2P – 1)(2P + 1) = 0 (mod q) What amounts to the same thing: Mp(Mp +2) = 0 (mod q) The stated conclusion now follows directly from Theorem 3.1. We cannot have both a|M, and q | Mp+2, for then q | 2, which is impossible. A single application should suffice to illustrate Theorem 11.3: if p = 23, then a = 2p +1 = 47 is also a prime, so that we may consider the case of M23. The question reduces to one of whether 47| M23 or, to put it differently, whether 225 = 1 (mod 47). Now, we have %3D 223 = 2 (2)* = 2°(-15)*(mod 47) %3D But (-15) = (225)? = (-10)² = 6 (mod 47) %3D Putting these two congruences together, we see that 223 = 23.6 = 48 = 1 (mod 47) whence M23 is composite. We might point out that Theorem 11.3 is of no help in testing the primality of M29, say; in this instance, 59 M29, but instead 59| M29 +2. or the two possibilities g| M, or g| Mp +2, is it reasonable to ask: What conditions on q will ensure that q | M,? The answer is to be found in Theorem 11.4.

Theorem 11.3. If p and q = 2p+1 are primes, then either q | Mp or q |Mp +2, but presented next. not both. Proof. With reference to Fermat's theorem, we know that 29-1 -1 = 0 (mod q) and, factoring the left-hand side, that (29-1)/2 - 1)(29-1)/2 + 1) = (2P – 1)(2P + 1) = 0 (mod q) What amounts to the same thing: Mp(Mp +2) = 0 (mod q) The stated conclusion now follows directly from Theorem 3.1. We cannot have both a|M, and q | Mp+2, for then q | 2, which is impossible. A single application should suffice to illustrate Theorem 11.3: if p = 23, then a = 2p +1 = 47 is also a prime, so that we may consider the case of M23. The question reduces to one of whether 47| M23 or, to put it differently, whether 225 = 1 (mod 47). Now, we have %3D 223 = 2 (2)* = 2°(-15)*(mod 47) %3D But (-15) = (225)? = (-10)² = 6 (mod 47) %3D Putting these two congruences together, we see that 223 = 23.6 = 48 = 1 (mod 47) whence M23 is composite. We might point out that Theorem 11.3 is of no help in testing the primality of M29, say; in this instance, 59 M29, but instead 59| M29 +2. or the two possibilities g| M, or g| Mp +2, is it reasonable to ask: What conditions on q will ensure that q | M,? The answer is to be found in Theorem 11.4.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 92E

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