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

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Discussion Overview

The discussion revolves around the relationship between prime pairs ending in 9 or 1 and their derivation from a specific mathematical form, namely 20x² - 1, as well as the exploration of properties related to centered decagonal primes. Participants engage in mathematical reasoning, conjectures about the infinitude of prime pairs, and the convergence of certain series.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that prime pairs derived from 20x² - 1 can only have factors ending in the digits 1 or 9.
  • One participant challenges others to prove the existence of an infinite set of such prime pairs.
  • Another participant introduces a series involving the prevalence of primes and questions whether it converges to a specific value.
  • Concerns are raised about the definition of the series, noting that some numbers included are not prime and do not end in 1 or 9.
  • There is a discussion about the implications of convergence on the finitude of prime pairs, with some arguing that convergence does not provide a definitive answer.
  • One participant mentions that the series is a variant of the Euler phi function and discusses the relationship between the numerators and denominators of certain fractions.
  • A participant shares a discovery about numbers of the form 5x² + 5x + 1 being centered decagonal numbers and questions their practical applications.

Areas of Agreement / Disagreement

Participants express differing views on the implications of convergence regarding the number of prime pairs. While some argue that convergence suggests a finite number, others contend that it can also occur with infinitely many pairs. The discussion remains unresolved regarding the exact nature of these relationships.

Contextual Notes

There are limitations in the definitions and assumptions made about the series and the properties of the prime pairs discussed. The mathematical steps leading to conclusions about convergence and finitude are not fully resolved.

Who May Find This Useful

This discussion may be of interest to those exploring number theory, particularly in relation to prime numbers, mathematical series, and the properties of specific types of primes such as centered decagonal primes.

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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