Give proof and make two questions related to the Theorem There are various methods for determining whether certain special types ofMersenne numbers are prime or composite. One such test is presented next.NUMBERS OF SPECIAL FORM229Theorem 11.3. If p and q = 2p+1 are primes, then either q |Mp or q| Mp+ 2, butnot 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)(2q-)/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 both9q |Mp and q | Mp +2, for then q | 2, which is impossible.A single application should suffice to illustrate Theorem 11.3: if p = 23, then9 = 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) = (-10)°= 6 (mod 47)Putting 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 ofM29, say; in this instance, 59{ M29, but instead 59| M29 + 2.Of the two possibilities q| M, or q| Mp+ 2, is it reasonable to ask: Whatconditions on q will ensure that a LM.? The answer is to be found in Theorem 11.4.Theorem 11.4. If g = 2n +l is prime, then we have the following:7 (mod 8)

Name: Give proof and make two questions related to the Theorem There are var
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Description: Give proof and make two questions related to the Theorem There are various methods for determining whether certain special types ofMersenne numbers are prime or composite. One such test is presented next.NUMBERS OF SPECIAL FORM229Theorem 11.3. If p a