Largest known twin primes and Sophie Germain primes

Indlekofer, Karl-Heinz; Járai, Antal

doi:10.1090/s0025-5718-99-01079-0

Cited by 15 publications

(14 citation statements)

References 8 publications

(9 reference statements)

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“…The number of generators of these groups are 8, 8 and 4, so we have altogether 20 remainders for which some power fulfills the condition in Corollary 1.4. So the density of primes p for which E K is not strongly semistable in characteristic char K = p is at least ≥ 2/3 (the remainders for which we do not know the answer are 1,2,4,5,8,16,25,27,29,30).…”

Section: Using Sophie Germain Primesmentioning

confidence: 99%

“…If we would know that there exists infinitely many Sophie Germain primes, then we could conclude that the density of primes for which the bundle Syz(X 2 , Y 2 , Z 2 ) is not strongly semistable on a Fermat curve can be arbitrarily high. The biggest Sophie Germain prime which I have found in the literature (see [4]) is h = 2375063906985 • 2 19380 − 1. There should be known results in analytic number theory which imply that the density of primes with non strongly semistable behaviour is arbitrarily high.…”

Section: Using Sophie Germain Primesmentioning

confidence: 99%

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On a Problem of Miyaoka

Brenner

2005

Number Fields and Function Fields—Two Parallel Worlds

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We give an example of a vector bundle E on a relative curve C → Spec Z such that the restriction to the generic fiber in characteristic zero is semistable but such that the restriction to positive characteristic p is not strongly semistable for infinitely many prime numbers p. Moreover, under the hypothesis that there exist infinitely many Sophie Germain primes, there are also examples such that the density of primes with non strongly semistable reduction is arbitrarily high.

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Section: Using Sophie Germain Primesmentioning

confidence: 99%

Section: Using Sophie Germain Primesmentioning

confidence: 99%

On a Problem of Miyaoka

Brenner

2005

Number Fields and Function Fields—Two Parallel Worlds

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“…Since our prospective purpose is to reach such results as Járai et al published in [8][9][10][11][12][13][14][15][16][17][18][19][20][21], apart from primality tests, we also have to implement some efficient sieving programs. In the following part of this paper the work described in [9] is called Cc22 − project.…”

Section: Prime Huntingmentioning

confidence: 99%

Prime Numbers

Farkas¹,

Kiss²,

Papatyi³

et al. 2020

eis

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"``Prime hunting" can be considered as a research area of computational number theory. Its goal is to find special combinations of integers and prove their primality. Four research groups, established by A. Járai between 1992 and 2014, published numerous world class scientific results. In this period, due to Járai's arithmetic routines fastest in the world, they reached the world record 19 times, namely found the largest known twin primes 9 times, Sophie Germain primes 7 times, a prime of the form n^4+1, a number which is simultaneously twin and Sophie Germain prime and the three largest known primes forming a Cunningham chain of length 3 of the first kind. When A. Járai retired, the investigations of this area were suspended. In the beginning of 2020 the research was reopened by G. Farkas at the newly-founded campus (Szombathely) of ELTE. The first signal success came in the end of May 2020. They proved the primality of the numbers which form the largest known Cunningham chains of length 2 of the 2nd kind. In this paper we report on a newly started prime hunting project with the aim of increasing our students' research activity.

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“…Hitherto sieving is the best known method for finding all prime numbers in a large interval (obviously, one has better algorithms when the primality of a single number is to be decided). The implementation of the sieve is mainly due to Járai, who used this technique together with his coauthors in their record prime searches like [6], or more recently [7].…”

Section: Description Of the Test Programmentioning

confidence: 99%

On the importance of cache tuning in a cache-aware algorithm: A case study

Burcsi

Kovács

2007

Computers & Mathematics with Applications

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In the present paper we describe and analyse a sieving algorithm for determining prime numbers. This external memory algorithm contains several parameters which are related to the sizes of the levels in the memory hierarchy. We examine how we should choose the values of these parameters in order to obtain an optimal running time. We compare the running times obtained by varying the parameters. We conclude that in this specific problem fine tuning pays off as we got a speed-up of almost 40%.

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Largest known twin primes and Sophie Germain primes

Abstract: Abstract. The numbers 242206083 · 2 38880 ± 1 are twin primes. The number p = 2375063906985 · 2 19380 − 1 is a Sophie Germain prime, i.e. p and 2p + 1 are both primes. For p = 4610194180515 · 2 5056 − 1, the numbers p, p + 2 and 2p + 1 are all primes.

Cited by 15 publications

References 8 publications

On a Problem of Miyaoka

On a Problem of Miyaoka

Prime Numbers

On the importance of cache tuning in a cache-aware algorithm: A case study

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