Prove Theorem 11.3 Of the two possibilities q | Mp or q| Mp + 2, is it reasonable to ask: WhatTheorem 11.4. If q = 2n +1 is prime, then we have the following:M9, say; in this instance, 59 M29, but instead 59 | M29 +2.Theorem 11.3. If p and q =2p+1 are primes, then either q| M, or q| Mp+2, butOne such test is presented next.types ofnot both.Proof. With reference to Fermat's theorem, we know that29-1-1=0 (mod q)and, factoring the left-hand side, that(24-1)/2 - 1)(24-1)/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 bothg|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, thenq = 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 2231 (mod 47). Now, we have223 = 2 (2) = 23(-15)*(mod 47)%3DBut(-15) = (225) = (-10)² = 6 (mod 47)Tuling these two congruences together, we see that223 = 23.6 = 48 = 1 (mod 47)whence M23 is composite.e might point out that Theorem 11.3 is of no help in testing the primality ofConditions on qwill ensure that g | M,? The answer is to be found in Theorem 11.4(a)9|M(b) alMprovided that qg = 1 (mod 8) or q = 7 (mod 8).O (mod 8) or a = 5 (mod 8).

Name: Prove Theorem 11.3 Of the two possibilities q | Mp or q| Mp + 2, is it
Uploaded: 2024-03-20T06:47:08.000Z
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Description: Prove Theorem 11.3 Of the two possibilities q | Mp or q| Mp + 2, is it reasonable to ask: WhatTheorem 11.4. If q = 2n +1 is prime, then we have the following:M9, say; in this instance, 59 M29, but instead 59 | M29 +2.Theorem 11.3. If p and q =2p+1 ar