Are Prime Pairs Ending in 9 or 1 Derived from 20x2-1 Centred 10-Gonal Primes?

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The discussion centers on the mathematical exploration of prime pairs derived from the formula 20x² - 1, specifically focusing on pairs ending in 9 or 1. Participants analyze the convergence of a series related to these primes, noting that while there are convincing arguments for the infinitude of such pairs, no definitive proof exists. The conversation also touches on the properties of centered decagonal primes and their relation to the Euler phi function, emphasizing the complexity of determining the number of prime pairs.

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Janosh89
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45x2+15x +/-1. ... 59;61 ,209;211 ,449;451 ,779;781 ...
45x2- 15x +/-1 ... 29;31 ,149;151 ,359;361 ,659;661 ...
Derived from 20x2-1 can only have factors ending in the digit _1, or _9 .
 
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Now prove that there is an infinite set of them ;).
 
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If I use S° as a symbol to mirror x,
The prevalence of primes in each strand ,+1 or -1 ,
Is S° *(9/11)*(17/19)*(27/29)*(29/31)*(39/41)... does this converge to some value?
 
It is strictly decreasing and bounded by 0. It will certainly converge.

9 and 27 are not prime and 17 does not end in 1 or 9, so I'm unsure how exactly your series is defined.
 
mfb said:
It is strictly decreasing and bounded by 0. It will certainly converge.

<Snip>.

More so if it is finite. Still, I guess this would imply there is a largest such pair. What other way is there to test for finitude of the pairs?
 
We cannot conclude how many pairs there are from the convergence.
The question how many prime pairs exist is an unsolved problem in mathematics - there are convincing arguments that their number is infinite, but there is no proof yet.
 
mfb said:
We cannot conclude how many pairs there are from the convergence.
The question how many prime pairs exist is an unsolved problem in mathematics - there are convincing arguments that their number is infinite, but there is no proof yet.
I know, I was going in the opposite direction: if there are finitely-many then the product converges.
 
Well, it also converges with infinitely many.
 
The series is a variant of the Euler phi function. Since the generating function (of x) contains quadratic (i.e. 2nd order)
terms, there are 9( or 11-2) terms that are not multiples of factor_11, 17(or 19-2) terms that are not 19n multiples,
27(or 29-2) that are not 29n multiples per 11,19,29 terms respectively.And so on.
9,17,27 are the numerators of the fractions with the denominator 11,19,29 respectively. The product of these fractions,
when multiplied by 11*19*29 gives 4131 (9*17*27) as the number or terms ,per 6061(11*19*29), that are relatively prime in respect to the factors 11,19,29.
 
  • #10
I have discovered in the last month that numbers of the form 5x2+5x+1, the core for expansion to 45x2 +15x+1, are Centred Decagonal Numbers
[@]prime-numbers.info
Do figurative prime numbers -like Decagonal Prime Numbers - have any use, apart from teaching and
possibly geometry?
 
  • #11
https://prime-numbers.info
scroll down to Prime Number Types#
will close this old thread for further replies -think this link URL will work
 
  • #12
Also called Centred 10-Gonal Primes:~
A062786
This also contains a "back of the envelope" -type proof re factors that can divide (divisors)
Note to staff: can this thread be blocked/ locked to further replies in 24/36 hours time?
 

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