Periodicity of the congruence 1666667 mod 666667
 

The numbers pg(k) are introduced:
pg(k) is the concatenation of two consecutive Mersenne numbers
pg(1)=10
pg(2)=31
pg(3)=73
pg(4)=157
.
.
.
using Julia software I searched the k's for which pg(k) is congruent to 1666667 mod 666667.
I saw that periodically there are three consecutive pg(k)'s congruent to 1666667 mod 666667.

Infact pg(18),pg(19) and pg(20) are congruent to 1666667 mod 666667...then pg(85196),pg(85197),pg(85198)...then pg(170374),pg(170375),pg(170376).
Is there a periodicity?

congruences
 

pg(255551),pg(255552) and pg(255553) are another triple congruent to 1666667 mod 666667

You mean numbers congruent to 333333 (mod 666667)


So what?

congruent to 6 mod 7
 

[QUOTE=LaurV;501306]You mean numbers congruent to 333333 (mod 666667)


So what?[/QUOTE]



I found about 40 primes of the form (2^k-1)*10^d+2^(k-1)-1 where d is the number of decimal digits of 2^(k-1)-1. None of these prime is congruent to 6 mod 7, whereas there are 9 primes of this form congruent to 5 mod 7. What do you think it is due?

No idea (I said this before).

There is no reason why these numbers wouldn't be 6 (mod 7).

Maybe it is due to [URL="https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers"]Guy's Law[/URL]?

What is the point of this thread?
 

Numbers of the form 13 mod 666 are my favourite. If only we could discover more of them then we might find some very important significance if we deeply investigate them. Then again, maybe some other random set of x mod y will yield a better chance of a major discovery? Than again again, maybe not. It's probably all just wasted characters on a webpage.

for a pair of consecutive Mersennes mod 7, to create 6 mod 7 we have:

(1,0) d=3 mod 6
(3,1) d=4 mod 6
(0,3) d does not exist.

[QUOTE=science_man_88;501333]for a pair of consecutive Mersennes mod 7, to create 6 mod 7 we have:

(1,0) d=3 mod 6
(3,1) d=4 mod 6
(0,3) d does not exist.[/QUOTE]




I just conjectured that there is no prime 6 mod 7 of the form (2^k-1)*10^d+2^(k-1)-1.
I think that up to k=800.000 there is no prime 6 mod 7.

Moreover the exponents leading to a prime seem to be NOT random at all...for example there is an exponent which is 51456...then an exponent which is 541456...note also that 541456-51456=700^2....
exponents of these primes are NOT random at all and residue 5 mod 7 occur twice than expected

i posted the question on mathoverflow and nobody yet has found an explanation

If they're not random, predict the next one.
Numerology is not evidence of non-randomness!