Periodicity of the congruence 1666667 mod 666667

enzocreti · 2018-11-26, 15:22

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?

enzocreti · 2018-11-26, 15:31

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

LaurV · 2018-11-30, 06:46

You mean numbers congruent to 333333 (mod 666667)

So what?

enzocreti · 2018-11-30, 12:14

Quote:

Originally Posted by LaurV

You mean numbers congruent to 333333 (mod 666667)

So what?

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?

LaurV · 2018-11-30, 15:46

No idea (I said this before).

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

Maybe it is due to Guy's Law?

retina · 2018-11-30, 16:19

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.

science_man_88 · 2018-11-30, 16:33

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.

enzocreti · 2018-11-30, 17:20

Quote:

Originally Posted by science_man_88

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.

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.

enzocreti · 2018-11-30, 17:25

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

enzocreti · 2018-11-30, 17:28

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

VBCurtis · 2018-11-30, 17:28

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

2018-11-26, 15:22	#1
enzocreti Mar 2018 2×5×53 Posts	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? Last fiddled with by enzocreti on 2018-11-26 at 15:26

2018-11-26, 15:31	#2
enzocreti Mar 2018 2·5·53 Posts	congruences pg(255551),pg(255552) and pg(255553) are another triple congruent to 1666667 mod 666667 Last fiddled with by enzocreti on 2018-11-26 at 16:28

2018-11-30, 16:19	#6
retina Undefined "The unspeakable one" Jun 2006 My evil lair 1834₁₆ Posts	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.

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2018-11-30, 06:46	#3
LaurV Romulan Interpreter Jun 2011 Thailand 7·1,373 Posts	You mean numbers congruent to 333333 (mod 666667) So what?

2018-11-30, 15:46	#5
LaurV Romulan Interpreter Jun 2011 Thailand 7·1,373 Posts	No idea (I said this before). There is no reason why these numbers wouldn't be 6 (mod 7). Maybe it is due to Guy's Law?

2018-11-30, 16:33	#7
science_man_88 "Forget I exist" Jul 2009 Dumbassville 10000011000000₂ Posts	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.

2018-11-30, 17:25	#9
enzocreti Mar 2018 2×5×53 Posts	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

2018-11-30, 17:28	#10
enzocreti Mar 2018 2·5·53 Posts	i posted the question on mathoverflow and nobody yet has found an explanation

2018-11-30, 17:28	#11
VBCurtis "Curtis" Feb 2005 Riverside, CA 11375₈ Posts	If they're not random, predict the next one. Numerology is not evidence of non-randomness!