The twin prime conjecture and other curiosities regarding prime numbers

Betti, Renato

doi:10.1007/s40329-017-0205-1

Cited by 2 publications

(3 citation statements)

References 5 publications

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“…Any two odd prime numbers in the sequences (1.1) differ by at least 2. Pairs of prime numbers that differ by 2 as, for example, in the sequence below (1.2) (3,5), (5,7), (11,13), (17,19), (29, 31), (41, 43), . .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Note that if p n is of the form 6y − 1 then 6(y + p n s) − 1, s > 0 is a multiple of p n in a pair in S and 6(p n − y + p n s) + 1, s ≥ 0 is a multiple of p n in a pair in S. Similarly if p n is of the form 6z + 1 then 6(z + p n s) + 1, s > 0 is a multiple of p n in a pair in S and 6(p n − z + p n s) − 1, s ≥ 0 is a multiple of p n in a pair in S. Thus the sieve is simultaneously defined on N and S and in S the residue set consists of twin primes. Thus π 2 (6k + 1) = |N * (E)| + 1, taking into account the pair (3,5).…”

Section: A Sieve Of Eratosthenes For Twin Primesmentioning

confidence: 99%

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Sieve methods and the twin prime conjecture

Mothebe¹

2019

Preprint

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Let p 1 = 2, p 2 = 3, p 3 = 5, . . . , p n , . . . be the ordered sequence of consecutive prime numbers in ascending order. For a positive integer m, denote by π(m) the number of prime numbers less than or equal to m. Let [ ] denote the floor or greatest integer function. In this paper, we show that for all n ≥ 1 :As a consequence, we see that there are infinitely many primes (Euclid's theorem). Then, we prove that if we let π 2 (m), denote the number of twin primes not exceeding m, then for all n ≥ 2 :and thereby prove the twin prime conjecture, namely that there are infinitely many prime numbers p such that p + 2 is also prime.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: A Sieve Of Eratosthenes For Twin Primesmentioning

confidence: 99%

Sieve methods and the twin prime conjecture

Mothebe¹

2019

Preprint

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“…The conjecture of twin primes as the 8th unsolved mathematical problem was proposed in the conference of the mathematicians over the world in 1900 years. During the more than one hundred years, it can be found from some mathematical journals that the conjecture of twin primes as an important topic has been attracting the exciting interests of a lot of researchers who work in the fields of mathematics over the world [2][3][4][5][6][7][8][9][10][11][12][13]. Up till now, rigorous and satisfactory proof on this conjecture of using the purely mathematical theories has not yet born in the world.…”

Section: Introductionmentioning

confidence: 99%

A Proof on the Conjecture of Twin Primes

Zhang¹

2019

IJAMTP

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Although the mathematicians all over the world offered hard explorations of more than one hundred years, the proof of using pure mathematical theories on the conjecture of twin primes has not born in the world. This paper is trying to apply computer program to prove that corresponding to infinite primes p, there are infinite p+2 primes. As a mathematical proof, the paper uses the concept of mapping to connect the computer program and the pure mathematical theory. With the requirement of a mathematical proof, in accord with the restriction of the integer of which the computer allows to take, an assumption is suggested, and on the basis of it, using the program of C language the paper presents, or regarding the C program as the mapping from infinite p primes to infinite p+2 primes, the paper proves that corresponding to infinite primes p, there are infinite p+2 primes; namely, the conjecture of twin primes is true.

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The twin prime conjecture and other curiosities regarding prime numbers

Cited by 2 publications

References 5 publications

Sieve methods and the twin prime conjecture

Sieve methods and the twin prime conjecture

A Proof on the Conjecture of Twin Primes

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