Assorted formulas for exponents of Mersenne primes

[Lee Yiyuan]

Originally Posted by S34960zz

As answered by Mini-Geek above, the exponent 131073 is divisible by 3.

This exponent can be checked easily, as its repeated sum-of-digits is divisible by 3.

1+3+1+0+7+3 = 15
1+5 = 6
6 is divisible by 3, thus 131073 is divisible by 3, thus M131073 is not prime.

See:
http://en.wikipedia.org/wiki/Divisibility_rule, and its sections:
http://en.wikipedia.org/wiki/Divisib...s_1.E2.80.9320
http://en.wikipedia.org/wiki/Divisib...isibility_by_3
http://en.wikipedia.org/wiki/Divisib...le#cite_note-0
The last link "cite_note-0" is an explanation of why the sum-of-digits method works for 3, and is the same mechanism used in Lucas-Lehmer testing.
(As a rank amateur in pure mathematics, I found this to be an interesting page. I have known about the divisibility rule for 3 since grade school, but not why it worked nor about the rules for the other numbers listed.)

See also:
http://primes.utm.edu/mersenne/index.html#known, referenced from
http://www.mersenne.org/ (under Results Queries in left margin).

Thank you for teaching me such a cool method to factorize numbers, it will be a very useful method for me!

[Lee Yiyuan]

is M536870923 a Mersenne prime??? Someone verify for me please my computer sucks

[Karl M Johnson]

Not prime

[science_man_88]

Originally Posted by Lee Yiyuan

is M536870923 a Mersenne prime??? Someone verify for me please my computer sucks

so far we know http://oeis.org/A000043 are the M(x) that are prime, with a few higher but unknown order ( not verified to be the nth Mersenne prime.

[axn]

Originally Posted by Lee Yiyuan

2) floor[lg(p + or - 1) / lg2] must = ceiling[lg(p + or - 1) / lg2]
3) ( 2^p + 1)/3 must not be evenly divisible by 3 or 43.

Weird. Never heard of these before. As far as we know, there are no such restrictions (In fact, most of the mersenne primes violate #2.)

[CRGreathouse]

Originally Posted by Lee Yiyuan

In order for 2^p - 1 to be prime,

1) p must be prime.
2) floor[lg(p + or - 1) / lg2] must = ceiling[lg(p + or - 1) / lg2]
3) ( 2^p + 1)/3 must not be evenly divisible by 3 or 43.

Is this correct/incorrect?

1 is correct. 2 and 3 are incorrect; 13 and 2 are counterexamples.