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Forum: Dobri 2022-04-19, 15:14
Replies: 11
Views: 1,760
Posted By Dobri
Addendum: A typo (an accidental inclusion of the...

Addendum: A typo (an accidental inclusion of the k term) has been made inadvertently in the previous post #11,
https://www.mersenneforum.org/showpost.php?p=604211&postcount=11.
Therefore, post #11...
Forum: Dobri 2022-04-18, 16:17
Replies: 11
Views: 1,760
Posted By Dobri
If xp-1+xp-2+...+x+1 = (x+p)(x+...)... over the...

If xp-1+xp-2+...+x+1 = (x+p)(x+...)... over the integers modulo 2kp+1,
then 2kp+1 is a factor of the composite Mersenne number Mp.

Also, if 2kp+1 is a factor of a composite Mersenne number Mp, ...
Forum: Dobri 2022-04-10, 14:28
Replies: 11
Views: 1,760
Posted By Dobri
Let p=1277, then the first integer k for which...

Let p=1277, then the first integer k for which 2kp+1 is a prime is k=10 so that 2×10×1277+1 = 25541.
Thus x1277-1+x1275+...+x+1 can be fully factored over the integers modulo 25541 as shown in the...
Forum: Dobri 2022-04-10, 13:41
Replies: 11
Views: 1,760
Posted By Dobri
Let n = p, then xp-1+...+1 = (x+p-1)p-1 over the...

Let n = p, then xp-1+...+1 = (x+p-1)p-1 over the integers modulo the prime number p.
For example:
x2-1+1 = (x+2-1)2-1 = x+1 over the integers modulo 2;
x3-1+x+1 = (x+3-1)3-1 = (x+2)2 over the...
Forum: Dobri 2022-04-10, 12:06
Replies: 11
Views: 1,760
Posted By Dobri
Addendum to post #1 (p = 11) for completeness: ...

Addendum to post #1 (p = 11) for completeness:
23 = 2×11+1 and 89 = 2×4×11+1
because the factors of composite Mersenne numbers are prime numbers n = 2kp+1.
Forum: Dobri 2022-04-10, 11:50
Replies: 11
Views: 1,760
Posted By Dobri
Let p = 13, then the corresponding Mersenne prime...

Let p = 13, then the corresponding Mersenne prime is M13 = 213-1 = 8191.
The polynomial x12+x11+...+x+1 can be fully factored over the integers modulo the prime numbers n = 2kp+1 as follows: ...
Forum: Dobri 2022-04-10, 06:53
Replies: 11
Views: 1,760
Posted By Dobri
It can be done, here is an example for p = 16: ...

It can be done, here is an example for p = 16:
(x16-1)/(x-1) = (1+x) (1+x2) (1+x4) (1+x8).
However, note that p = 16 is a composite number.
Forum: Dobri 2022-04-10, 06:29
Replies: 11
Views: 1,760
Posted By Dobri
Concerning the Galois fields GF(n), n has to be...

Concerning the Galois fields GF(n), n has to be either a prime number or a prime power.
Forum: Dobri 2022-04-10, 06:09
Replies: 11
Views: 1,760
Posted By Dobri
Factorization of (x^p-1)/(x-1) over the integers modulo n

Let's start this thread with the polynomial (x^p-1)/(x-1) for p = 11, so that
(x11-1)/(x-1) = x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1.
The corresponding composite Mersenne number is M11 = 211-1 = 2047 = 23...
Showing results 1 to 9 of 9

 
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