541456 and 51456. I checked 20 numbers 2000 times and found 200 patterns!!
pg(k)=M(k)M(k1) where  denotes concatenation base 10 and M(k) is the kth Mersenne number. Examples of such numbers are 1023511, 255127, 12763...
I found that pg(k) is prime for k=51456 and for k=541456. I wonder if that is only a coincidence (54145651456=700^2) or there is an hidden structure undermeath these numbers. What is the probability to have by mere chance two probable primes with these two amazing exponents? 
[QUOTE=enzocreti;502374]I wonder if that is only a coincidence[/quote]
Yes, unfortunately. [QUOTE=enzocreti;502374]What is the probability to have by mere chance two probable primes with these two amazing exponents?[/QUOTE] This question is not very meaningful. You haven't specified what property(ies) of this pair of numbers makes them "amazing". Once you specify that in a more mathematically rigorous way, we can attempt to quantify the probability. As such, what you're doing is looking for afterthefact coincidences. When you started out with this series, you were only focusing on the lack of 6 (mod 7) primes. Now you're focusing on pairs of "amazing" exponents. But none of these were predicted beforehand. You just looked at the numbers (which gives primes) and tried to find coincidences, and you did. Nothing more. 
IT IS NOT A COINCIDENCE!
[QUOTE=axn;502376]Yes, unfortunately.
This question is not very meaningful. You haven't specified what property(ies) of this pair of numbers makes them "amazing". Once you specify that in a more mathematically rigorous way, we can attempt to quantify the probability. As such, what you're doing is looking for afterthefact coincidences. When you started out with this series, you were only focusing on the lack of 6 (mod 7) primes. Now you're focusing on pairs of "amazing" exponents. But none of these were predicted beforehand. You just looked at the numbers (which gives primes) and tried to find coincidences, and you did. Nothing more.[/QUOTE] I am sure it is NOT a coincidence!!! 
pg(2131) and pg(2131*9=19179)
also this is a coincidence: pg(2131) is prime and pg(2131*9) is prime!!!

[QUOTE=enzocreti;502378]I am sure it is NOT a coincidence!!![/QUOTE]
[QUOTE=enzocreti;502380]also this is a coincidence: pg(2131) is prime and pg(2131*9) is prime!!![/QUOTE] I understand that you think these are not coincidences. You will keep on trying to find "hidden patterns". I have said all I can. Good luck. 
[QUOTE=enzocreti;502380]also this is a coincidence: pg(2131) is prime and pg(2131*9) is prime!!![/QUOTE]
Yes, you're correct here this is a coincidence. You have no idea what you mean what you say "this is not a coincidence," and it makes you look like a fool. You're practicing the primenumber version of throwing darts at a dartboard, and then expressing amazement that you can find some meaning in the score. Perhaps you have a knack for astrology; those people find meaning in trivial happenings too. 
[QUOTE=enzocreti;501344]2,3,4,7,8,12,19,22,36,46,51,67,79,215,359,394,451,1323,2131,3336,3371,6231,19179=9*2131,39699,51456,56238,69660,75894,79798,92020,174968, 176006,181015,285019,331259,360787,366770,...,541456[/QUOTE]
[QUOTE=enzocreti;502380]also this is a coincidence: pg(2131) is prime and pg(2131*9) is prime!!![/QUOTE] I think you might be on to something. pg(36) is prime, and so is pg(36*1935). pg(67) is prime, and so is pg(67*93). pg(67) is prime, and so is pg(67*768). pg(215) is prime, and so is pg(215*324). pg(215) is prime, and so is pg(215*428). I discarded the numbers under 36 (to avoid getting easy small multiples) and 541456 (because it wasn't clear if some numbers were skipped). This is far more than the number of multiples expected by chance. Does this suggest that multiples of earlier terms are more likely, or just that having small factors are more likely? It's not clear to me. 
[QUOTE=enzocreti;502374]... or there is an hidden structure undermeath these numbers?[/QUOTE]
Of course there is! This is what happens when you take off the tinfoil hat  the whole world starts bursting at its seams. Put it back on  immediately! [YOUTUBE]ShdmErv5jvs[/YOUTUBE] 
[QUOTE=CRGreathouse;502412]
pg(36) is prime, and so is pg(36*1935). pg(67) is prime, and so is pg(67*93). pg(67) is prime, and so is pg(67*768). pg(215) is prime, and so is pg(215*324). pg(215) is prime, and so is pg(215*428). [/QUOTE] are all multiples that work for larger values multiples of 3? 
Primes
[QUOTE=CRGreathouse;502412]I think you might be on to something.
pg(36) is prime, and so is pg(36*1935). pg(67) is prime, and so is pg(67*93). pg(67) is prime, and so is pg(67*768). pg(215) is prime, and so is pg(215*324). pg(215) is prime, and so is pg(215*428). I discarded the numbers under 36 (to avoid getting easy small multiples) and 541456 (because it wasn't clear if some numbers were skipped). This is far more than the number of multiples expected by chance. Does this suggest that multiples of earlier terms are more likely, or just that having small factors are more likely? It's not clear to me.[/QUOTE] I don't know, the only thing I can see is that these numbers are not random at all! 
WHO ARE YOU?
[QUOTE=VBCurtis;502410]Yes, you're correct here this is a coincidence.
You have no idea what you mean what you say "this is not a coincidence," and it makes you look like a fool. You're practicing the primenumber version of throwing darts at a dartboard, and then expressing amazement that you can find some meaning in the score. Perhaps you have a knack for astrology; those people find meaning in trivial happenings too.[/QUOTE] WHO ARE YOU? 
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