Theorem 11.3. If p and q = 2p+1: primes, then either q | M, or q| M,+2, but are not both.

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Chapter2: Second-order Linear Odes
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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 that
Theorem 11.3. If p and q = 2p+1 are primes, then either q | M, or q | M,+2, but
Whether certain special types of
not 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 both
a|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. The
question reduces to one of whether 47| M23 or, to put it differently, whether 223 =
1 (mod 47). Now, we have
223 = 2°(2°)* = 2°(-15)*(mod 47)
But
(-15)* = (225)? = (-10)² = 6 (mod 47)
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.
Of the two possibilities q| M, or q| Mp+2, is it reasonable to ask: What
found 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)
Transcribed Image Text:Mersenne numbers are prime or composite. One such test is presented next. Proof. With reference to Fermat's theorem, we know that Theorem 11.3. If p and q = 2p+1 are primes, then either q | M, or q | M,+2, but Whether certain special types of not 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 both a|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. The question reduces to one of whether 47| M23 or, to put it differently, whether 223 = 1 (mod 47). Now, we have 223 = 2°(2°)* = 2°(-15)*(mod 47) But (-15)* = (225)? = (-10)² = 6 (mod 47) 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. Of the two possibilities q| M, or q| Mp+2, is it reasonable to ask: What found 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)
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