Mersenne Numbers Known from Number Practice to Be Composite

[Dobri]

It should be noted that it is assumed that 6p + 1 is a prime number.
If 6p + 1 is not a prime number, there are instances when M_p mod (6p + 1) ≠ 0 and 6p + 1 = 27a² + b², for example:
p = 41, (2⁴¹ - 1) mod (6×41 + 1) = 31 ≠ 0, and 6×41 + 1 = 27×3² +2².

[kijinSeija]

Yes of course 6*p+1 must be prime I guess but p isn't necessary prime.

For example : (6*21+1) = 127 divides (2^21-1) but 21 isn't prime.

[Dobri]

Originally Posted by kijinSeija

Yes of course 6*p+1 must be prime I guess but p isn't necessary prime.

For example : (6*21+1) = 127 divides (2^21-1) but 21 isn't prime.

This forms a sequence of composite p values indeed,
p = 21, 121, 153, 221, 237, 245, 305, 333, 357, 381, 445, 465, 545, 565, 605, 637, 657, 737, 753, 777, 793, 861, 917,...

kc = 3; ic = 2; While[ic <= 1000, p = ic; Mn = 2^p - 1; fc = 2*kc*p + 1;
 If[(Mod[p, 4] == 1) && (Mod[Mn, fc] == 0) && (PrimeQ[p] == False) && (PrimeQ[fc] == True), 
  rc = FindInstance[{fc == 27*ac^2 + bc^2}, {ac, bc}, PositiveIntegers];
  Print[p, " ", Mod[Mn, fc], " ", rc];];

21 0 {{ac->1,bc->10}}
121 0 {{ac->3,bc->22}}
153 0 {{ac->3,bc->26}}
221 0 {{ac->7,bc->2}}
237 0 {{ac->7,bc->10}}
245 0 {{ac->1,bc->38}}
305 0 {{ac->5,bc->34}}
333 0 {{ac->7,bc->26}}
357 0 {{ac->1,bc->46}}
381 0 {{ac->9,bc->10}}
445 0 {{ac->9,bc->22}}
465 0 {{ac->5,bc->46}}
545 0 {{ac->11,bc->2}}
565 0 {{ac->1,bc->58}}
605 0 {{ac->9,bc->38}}
637 0 {{ac->7,bc->50}}
657 0 {{ac->11,bc->26}}
737 0 {{ac->11,bc->34}}
753 0 {{ac->5,bc->62}}
777 0 {{ac->13,bc->10}}
793 0 {{ac->13,bc->14}}
861 0 {{ac->7,bc->62}}
917 0 {{ac->1,bc->74}}
...

[Dr Sardonicus]

The OP to this thread indicates that we're only interested in prime exponents p. (After all, Mersenne numbers with composite exponents have algebraic factorizations, so are already "known from number theory to be composite.")

It has long been known that if p = 4*n + 3 and q = 2*p + 1 = 8*n + 7 are both prime, then q divides 2^p - 1.

This is because the quadratic character of 2 (mod q) is determined by q (mod 8).

For given k > 1, there is alas no similarly convenient criterion for when p and q = 2*k+p + 1 both being prime implies q divides 2^p - 1.

Since p = (q-1)/2/k, we can say that 2^((q-1)/2) == 1 (mod q), so that q == 1 or 7 (mod 8). This condition excludes k with k%4 = 2 from consideration.

Suppose that k > 1, k%4 <> 2, p is prime, and q = 2*k*p + 1 is also prime. We can choose p%4 to insure that q%8 is 1 or 7:

• If k%4 = 0, p == 1 or 3 (mod 4)
• If k == 1 (mod 4) then p == 3 (mod 4)
• If k == 3 (mod 4) then p == 1 (mod 4).

AFAIK the fastest way to determine whether q = 2*k*p + 1 divides 2^p - 1 for k > 1 is to test whether Mod(2,q)^p == 1.

I can confidently predict that q = 2*k*p + 1 will divide 2^p - 1 for about 1/k of the primes p for which q is also prime, if p%4 is chosen so that 2 is a quadratic residue (mod q). I just can't predict which 1/k it will be. Here is an actual count for p < 10^8:

? c1=0;c2=0;forprime(p=3,100000000,if(p%4==3,q=10*p+1;if(isprime(q),c1++;if(Mod(2,q)^p==1,c2++))));print(c1" "c2)
259346 51816
?

The proportion of primes p == 3 (mod 4) such that q = 10*p + 1 is also prime, for which q also divides 2^p - 1, is 0.19979+ which is pretty close to 1/5.

[Dobri]

Excluding the 3^rd Mersenne prime number M₅ (for which p = 5, p mod 4 = 1, 2^p - 1 = 6p + 1 = 27×1² + 2² = 31 is a prime, and also 2p + 1 = 11 is a prime):

1) Out of the total of 50,847,534 prime numbers 2 ≤ p ≤ 999,999,937, there are 1,046,030 prime exponents (2.057%) with composite Mersenne numbers M_p mod (6p + 1) = 0 for which p mod 4 = 1 and 6p + 1 = 27a² + b² is a prime.

2) Out of the said 1,046,030 prime exponents of composite Mersenne numbers, there are 55,953 prime exponents for which 2p + 1 is also a prime.

Note: The conjecture in post #17 (https://www.mersenneforum.org/showpo...4&postcount=17) holds in the interval of prime exponents 2 ≤ p ≤ 999,999,937.

Also, out of the total of 50,847,534 prime numbers 2 ≤ p ≤ 999,999,937, there are 3,138,462 prime numbers (p = 5 included) for which 6p + 1 is a prime.
Here 3,138,462 = 3 × 1,046,031 (p = 5 included) + 369.
Out of the said 3,138,462 prime numbers, there are 168,529 prime numbers (p = 5 included) for which 2p + 1 is also a prime.
Here 168,529 = 3 × 55,954 (p = 5 included) + 667.

[Dobri]

For k = 5, out of the total of 50,847,534 prime numbers 2 ≤ p ≤ 999,999,937, there are 408,660 prime exponents (0.8%) with composite Mersenne numbers M_p mod (10p + 1) = 0 for which p mod 4 = 3 and p mod 6 = 1.

Also, out of the total of 50,847,534 prime numbers 2 ≤ p ≤ 999,999,937, there are 4,084,314 prime numbers for which 10p + 1 is a prime.
Here 4,084,314 = 2,043,235 (for which p mod 4 = 1) + 2,041,079 (for which p mod 4 = 3),
4,084,314 = 4,084,313 (for which p mod 6 = 1) + 1 (for which p = 3, 2³ - 1 < 10×3 + 1), and
2,041,079 = 5 × 408,660 - 2,221.

[Dobri]

Number of 2kp+1 primes for p within tne range 2 ≤ p ≤ 999,999,937 which divide the composite Mersenne numbers M_p = 2^p-1 so that M_p mod (2kp+1) = 0:

k=1: 1655600 (for which p mod 4 = 3 and p mod 6 = 5);

k=2: None;

k=3: 1046030 (for which p mod 4 = 1 and 2×3p+1 = 27a²+b² is a prime);

k=4: 773708 (for which p mod 6 = 5), however, 54990 primes are already counted for k=1 and k=3 as follows: for 41786 primes M_p mod (2p+1) = 0, and for 13204 primes M_p mod (6p+1) = 0;

k=5: 408660 (for which p mod 4 = 3 and p mod 6 = 1);

k=6: None;

k=7: 258602 (for which p mod 4 = 1 and p mod 6 = 5), however, 15901 primes are already counted for k=3 and k=4 as follows: for 9327 primes M_p mod (6p+1) = 0, for 6751 primes M_p mod (8p+1) = 0, and for 9327+6751-15901=177 primes M_p mod (6p+1) = 0 and M_p mod (8p+1) = 0;
...

[Dobri]

This is a continuation of the previous post #29 at https://www.mersenneforum.org/showpo...4&postcount=29.
…
Number of 2kp+1 primes for p within the range 2 ≤ p ≤ 999,999,937 which divide the composite Mersenne numbers M_p = 2^p-1 so that M_p mod (2kp+1) = 0:

k=8: 375441 (for which p mod 6 = 1), however, 15228 primes are already counted for k=3 and k=5 as follows (note that k=8=3+5): for 9615 primes M_p mod (6p+1) = 0, and for 5613 primes M_p mod (10p+1) = 0;

k=9: 331102 (for which p mod 4 = 3), however, 30369 primes are already counted for k=1,4,5 and 8 as follows (note that k=9=1+8=4+5): for 17824 primes M_p mod (2p+1) = 0, for 6240 primes M_p mod (8p+1) = 0, for 4931 primes M_p mod (10p+1) = 0, and for 2002 primes M_p mod (16p+1) = 0, so that 17824+6240+4931+2002-30369=628 Mersenne numbers have more than two factors;

k=10: None;

k=11: 149424 (for which p mod 4 = 1 and p mod 6 = 1), however, 6898 primes are already counted for k=3 and k=8 as follows (note that k=11=3+8): for 5127 primes M_p mod (6p+1) = 0, and for 1854 primes M_p mod (16p+1) = 0, so that 5127+1854-6898=83 Mersenne numbers have more than two factors;
...

[kijinSeija]

Hi,

I noticed something about the prime of the form p and 8p+1 and their divisibility by 2^p-1

If p = ((2a-1)^2+64(2b-1)^2-1)/8 and 8p+1 = (2a-1)^2+64(2b-1)^2, then 8p+1 divides 2^p-1

To find the exponent I use this code on PARI GP :

for(a=1, 1000, for(b=1, 1000, if(isprime((((2*a-1)^2+64*(2*b-1)^2-1)/8))&&isprime((2*a-1)^2+64*(2*b-1)^2), print(((2*a-1)^2+64*(2*b-1)^2-1)/8))))

But I don't how to sort by increasing order on PARI GP

I don't know if it's useful and I didn't found a counterexample but this interesting I guess

[LaurV]

Originally Posted by kijinSeija

I noticed something about the prime of the form p and 8p+1 and their divisibility by 2^p-1

It is actually viceversa , 2^p-1 is divisible by 8p+1.

And yes, this is an interesting find, I was not aware of. Up to now, we don't have a result that says anything about small factors of mersenne for exponents p which are prime and 1 (mod 4). For p=3 (mod 4) we have the SG result (if q=2p+1 is prime and p is 3 mod 4, than q divides m=2^p-1). But there is no similar result known to me which is touching prime p which are 1 (mod 4). If the "trick" behind of those formulas (which I didn't check yet) can be proved, this is an interesting result, with some mild theoretic value.

And no, this is not "useful" in practice, there is no practical value, the factors it uncovers are very small, only 8 times the value of p, which are eliminated by initial sieving/TF in the first millisecond (for k=4, in q=2kp+1).

Related to "sorting" in pari, you can add all the things to a list and use the sorting functions, if you need, or put everything in a file and sort it externally, but this effort is not really justified if you observe that the part coming from your "b" is much higher than the part from "a", so if you move b to external loop, they come out pretty sorted...

for(b=1, 100, for(a=1, 100, q=(2*a-1)^2+64*(2*b-1)^2; p=(q-1)/8; if(isprime(p)&&isprime(q), print(q", divides 2^"p"-1"))))

(I also fixed a bit your calculation so you don't need to waste time calculating the same thing over and over, and also print both q and p. Note that there is no insurance of the fact that this is true, unless proved. You can add the "&&Mod(2,q)^p=1" in that "if" condition, and check if the resulted list is the same.

Edit:

Ok. Using these 3 lines, one by one, in order, we get:

gp> cnt=0; for(b=1, 1000, for(a=1, 1000, q=(2*a-1)^2+64*(2*b-1)^2; p=(q-1)/8; if(isprime(p)&&isprime(q), cnt++; print(cnt": "q", "p))))
(6421 results are printed)
gp> cnt=0; for(b=1, 1000, for(a=1, 1000, q=(2*a-1)^2+64*(2*b-1)^2;  p=(q-1)/8; if(isprime(p)&&isprime(q)&&Mod(2,q)^p==1, cnt++; print(cnt": "q", "p))))
(6421 results are printed)

gp> cnt=0; for(b=1, 1000, for(a=1, 1000, q=(2*a-1)^2+64*(2*b-1)^2;  p=(q-1)/8; if(isprime(p)&&isprime(q)&&Mod(2,q)^p==1&&Mod(p,4)==1, cnt++; print(cnt": "q", "p))))
(3420 results are printed)

Buddy, you are onto something...

[charybdis]

Originally Posted by kijinSeija

Hi,

I noticed something about the prime of the form p and 8p+1 and their divisibility by 2^p-1

If p = ((2a-1)^2+64(2b-1)^2-1)/8 and 8p+1 = (2a-1)^2+64(2b-1)^2, then 8p+1 divides 2^p-1

This is equivalent to saying that if 8p+1 is prime and of the form (2a-1)^2+64(2b-1)^2, then 2 is an 8th power mod 8p+1.

You've rediscovered an old result on when 2 is an 8th power modulo a prime. Reuschle (1856) stated, and Western (1911) proved, that if q is a prime of the form 8k+1:
- If k is even, then 2 is an 8th power mod q if and only if q is of the form x^2 + 256y^2
- If k is odd, then 2 is an 8th power mod q if and only if q is of the form x^2 + 64y^2 but not of the form x^2 + 256y^2.

Here we are in the k odd case, and your (2a-1)^2+64(2b-1)^2 exactly corresponds to saying that 8p+1 is of the form x^2 + 64y^2 but not of the form x^2 + 256y^2. So in fact we can strengthen your statement to an "if and only if".