Proof short and simplified with clear letters Mersenne numbers are prime or composite. One such test is presented next.Proof. With reference to Fermat's theorem, we know thatTheorem 11.3. If p and q = 2p+1 are primes, then either q | M, or q | M,+2, butWhether certain special types ofnot both.29-1 -1=0 (mod q)and, factoring the left-hand side, that(2(a-1)/2 – 1)(24-1)/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 botha|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, then=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°)* = 2°(-15)*(mod 47)But(-15)* = (225)? = (-10)² = 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: Whatfound in Theorem 11.4.Theorem 11.4. If q = 2n + 1 is prime, then we have the following:ta) q| M, provided that q = 1 (mod 8) or q = 7 (mod 8).0) 9|M,+ 2, provided that q = 3 (mod 8) or q = 5 (mod 8).24-1/2= 2" = 1 (mod q)

Name: Proof short and simplified with clear letters Mersenne numbers are pri
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Proof short and simplified with clear letters Mersenne numbers are prime or composite. One such test is presented next.Proof. With reference to Fermat's theorem, we know thatTheorem 11.3. If p and q = 2p+1 are primes, then either q | M, or q | M,+2,