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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Create and solve two assignments that represent the theorem 11.3
Transcribed Image Text:(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 i
Theorem 11.3. If p and q = 2p +1 are primes, then either q | M, or q | Mp+2, but
not both.
Bnof 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(M,+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
0= 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°)ª = 2°(-15)*(mod 47)
But
(-15)* = (225)² = (-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
29, 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: What
nuitions 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:
%3D
91M provided that q1 (mod 8) or q 7 (mod 8).
Proof, To
aouivalent to asserting that
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