Assorted formulas for exponents of Mersenne primes

[Lee Yiyuan]

My Java Netbeans says that M131073 is prime. Someone verify for me please.

[TimSorbet]

Originally Posted by Lee Yiyuan

My Java Netbeans says that M131073 is prime. Someone verify for me please.

131073=3*43691

[Lee Yiyuan]

Originally Posted by Mini-Geek

131073=3*43691

Thank you.

[Lee Yiyuan]

i recently saw this article:

http://en.wikipedia.org/wiki/Mersenne_conjectures

Can it be somehow utilized?

[science_man_88]

CRG with what you know of Me(x) function what are the odds of this :

(12:21)>for(x=1,39,print1((Me(x)^x^2-1)%4))
100000200020002000000000000020200000000
(12:22)>for(x=1,39,print1((Me(x)^x^3-1)%4))
100000200020002000000000000020200000000
(12:22)>for(x=1,39,print1((Me(x)^x^4-1)%4))
100000200020002000000000000020200000000
(12:23)>for(x=1,39,print1((Me(x)^x^5-1)%4))
1000002000200020000000000000202

for anyone else Me is a Mersenne exponent function.

[10metreh]

Originally Posted by science_man_88

according to number freak the formula is f(x)=x^2+x+41 so f(40)= 40^2+40+41 = 1600 + 81 = 1681.

However, the formula in mart_r's post is 41+n(n-1) which is effectively the same sequence because n^2+n+41 = n(n+1)+41, but f(n) in mart_r's version is equal to f(n-1) in Euler's version. This explains the difference.

[science_man_88]

Originally Posted by 10metreh

However, the formula in mart_r's post is 41+n(n-1) which is effectively the same sequence because n^2+n+41 = n(n+1)+41, but f(n) in mart_r's version is equal to f(n-1) in Euler's version. This explains the difference.

while you're having fun with that care to comment on any of my latest ideas ?

[Uncwilly]

Originally Posted by Lee Yiyuan

My Java Netbeans says that M131073 is prime. Someone verify for me please.

The exponent has to be prime before the Mersenne number can be prime. It is not even worth checking them otherwise.

[wblipp]

Originally Posted by Lee Yiyuan

My Java Netbeans says that M131073 is prime. Someone verify for me please.

Originally Posted by Uncwilly

The exponent has to be prime before the Mersenne number can be prime. It is not even worth checking them otherwise.

In greater detail, because 131073 = 3 * 43691, we know that M131073 is divisible by M3 and M43691.

[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?

[S34960zz]

Originally Posted by Lee Yiyuan

My Java Netbeans says that M131073 is prime. Someone verify for me please.

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).