Program for Sage
 

pg(k) numbers are so defined:


pg(k)=(2^k-1)*10^d+2^(k-1)-1 where d is the number of decimal digits of 2^(k-1)-1.
The numbers are formed by the concatenation base 10 of two consecutive Mersenne numbers, examples are 157, 40952047.
I conjectured that there is no prime of this form congruent to 6 modulo 7.
has somebody an efficient routine for Sage for testing this conjecture?

Thread moved to blog area

[QUOTE=enzocreti;501564]
I conjectured that there is no prime of this form congruent to 6 modulo 7.
[/QUOTE]
It has been explained to you in other threads that this 'conjecture' is almost obviously false.
It is similar to a 'conjecture' that there are no 2-million-digit numbers, because 'I cannot imagine them and I cannot test them. Will someone test them for me?'. That's not a conjecture - that's a limitation of means.

Probable primes 5 and 6 mod 7
 

I found 9 probable primes with residue 5 mod 7 and none with residue 6 mod 7 (residue 5 and 6 occur with the same frequency). That cannot be coincidence!

[QUOTE=enzocreti;501593]That cannot be coincidence![/QUOTE]

Correlation != Causality.

[QUOTE=enzocreti;501593]I found 9 probable primes with residue 5 mod 7 and none with residue 6 mod 7 (residue 5 and 6 occur with the same frequency). That cannot be coincidence![/QUOTE]

primes other than 7 can be 1,2,3,4,5,6 mod 7 the odds in theory of a prime not being 6 mod 7 given equal probability ( a potentially invalid assumption) is 5/6 so the odds that 5 primes aren't 6 mod 7 is (5/6)^5 = (5^5)/(6^5) = 3125/7776 ~40% still not that bad. 5 mod 7 are also 19 mod 42, 6 mod 7 are all 13 mod 42. even with 9 it is still roughly 20%

[QUOTE=science_man_88;501598]primes other than 7 can be 1,2,3,4,5,6 mod 7 the odds in theory of a prime not being 6 mod 7 given equal probability ( a potentially invalid assumption) is 5/6 so the odds that 5 primes aren't 6 mod 7 is (5/6)^5 = (5^5)/(6^5) = 3125/7776 ~40% still not that bad. 5 mod 7 are also 19 mod 42, 6 mod 7 are all 13 mod 42. even with 9 it is still roughly 20%[/QUOTE]

ok and what about the "coincidence" of an exponent leading to a prime which is 51456 and another exponent leading to a prime which is 541456...where 541456-51456=700^2. These numbers have an hidden structure!

Another "coincidence": an exponent leading to a prime is 2131, another is 19179=2131*9, to me seems quite clear that these numbers have some hidden structure...it is too difficult for me to say what is this hidden structure but it MUST exist!

primes
 

Maybe a specialist in Number theory knows something about these primes...i repeat look at the exponents...215,92020,69660...all multiples of 215...it is too hard anyway for me to find a pattern

255127
 

One of these primes 255127 divides 2^258+3.

[QUOTE=enzocreti;501604]Maybe a specialist in Number theory knows something about these primes...i repeat look at the exponents...215,92020,69660...all multiples of 215...it is too hard anyway for me to find a pattern[/QUOTE]

you can use the form in a possible proof of the claim ... : [url]https://en.m.wikipedia.org/wiki/Mathematical_proof[/url]