# There are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31

Abstract : We proved that $\liminf\limits_{n \rightarrow +\infty}(p_{n+1}-p_n)=2$ where $p_n$ is the $n-th$ prime number. We showed the conditions on which an integer $10X+1$, $10X+3$, $10X+7$ or $10X+9$ can be a prime number. We studied the conditions required to get some twin primes and proved the twin primes conjecture. First we showed that some sets of formulas are governing each of the eight sets of composite integers 30n+11, 30n+31, 30n+13, 30n+23, 30n+7, 30n+17, 30n+19 and 30n+29. Then we showed that there are infinitely many couple of integers 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31 which are not generated by the formulas of the corresponding sets of composite integers respectively. As a side result, it will open a new era for the study of gaps between primes and leads to the proofs of many open problems linked to prime numbers.
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https://hal.archives-ouvertes.fr/hal-01307789
Contributor : Sibiri Christian Bandre <>
Submitted on : Tuesday, May 3, 2016 - 7:00:42 PM
Last modification on : Wednesday, May 11, 2016 - 1:01:53 AM
Long-term archiving on : Tuesday, May 24, 2016 - 4:31:12 PM

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• HAL Id : hal-01307789, version 1

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Sibiri Christian Bandre. There are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31. 2016. ⟨hal-01307789⟩

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