Largest Known Twin Primes

Parady, B. K.; Smith, Joel F.; Zarantonello, Sergio E.

doi:10.2307/2008812

Cited by 7 publications

(9 citation statements)

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“…. .. To our best knowledge the earlier records were 1692923232•10 4020 ±1 (4030 digit), 4655478828•10 3429 ±1 (3439 digit) found by Dubner [1] and 1706595 • 2 11235 ± 1 (3389 digit) found by Parady, Smith, Zarantonello [3] in 1990. The search consisted of the following steps:…”

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confidence: 72%

Largest known twin primes

Indlekofer¹,

Járai²

1996

Math. Comp.

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Abstract. The numbers 697053813 · 2 16352 ± 1 are twin primes.A search for large prime pairs of the form (3 + 30h) · 2 16352 ± 1 was performed at the University of Paderborn, Germany in the framework of a project for parallel computing in computational number theory supported by the Heinz Nixdorf Institute, Paderborn. We used the arithmetic routines developed in this project for fast parallel and sequential computations with very large numbers. The search was done until the first large prime pair 697053813 · 2 16352 ± 1 was found. 2. Both the case +1 and the case −1 were sieved with factors from 7 up to 2 37 . After sieving, 209571 candidates remained.3. Using the probabilistic Miller-Rabin primality test (see Knuth [2,), the candidates were tested, first the +1 case and then the −1 case, until a "probable twin prime pair" was found.4. This "probable twin prime pair" was tested with exact tests: the −1 case using a Lucasian test (see Riesel [5]) and the +1 case using the test of Brillhart, Lehmer and Selfridge and the test of Proth (see Ribenboim [4,).

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confidence: 72%

Largest known twin primes

Indlekofer¹,

Járai²

1996

Math. Comp.

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“…The twin primes conjecture asserts that the gap g = 2 occurs infinitely often. Work on this conjecture has included computer-based enumerations [9,11,13] and investigations of Brun's constant [4,16,6,14]. Brun's constant is the sum of the reciprocals of twin primes.…”

Section: Related Resultsmentioning

confidence: 99%

“…Of the last 13 copies of s = 66 in G(11 # ) none survive the recursion; each is subject to at least one addition as the recursion continues. In the interval [13,2310] covered by G(11 # ), fifteen additional copies of s = 66 are created from the driving terms of s = 66 (such as s = 624).…”

Section: Figure 2 This Graph Depicts the Percentage Error Between Our...mentioning

confidence: 99%

Estimating constellations among primes - I. Uniformity

Holt,

Rudd

2013

Preprint

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A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations.In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression.For these initial estimates, we assume that the copies of a given constellation tend toward a uniform distribution in the cycle of gaps, as the recursion progresses. Our simple estimates based on the recursion of gaps and the assumption of uniformity appear to have correct asymptotic behavior, and they exhibit a systematic error correlated to length of the constellation.

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“…The twin primes conjecture asserts that the gap g = 2 occurs infinitely often. Work on this conjecture has included computer-based enumerations [12,15,18] and investigations of Brun's constant [8,22,10,20]. Brun's constant is the sum of the reciprocals of twin primes.…”

Section: Related Resultsmentioning

confidence: 99%

Expected gaps between prime numbers

Holt

2007

Preprint

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We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erdös and Turán.

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Largest Known Twin Primes

Abstract: Abstract.663777 • 27650 ± 1 , 571305 • 27701 ± 1 and 1706595 • 211235 ± 1 are twin primes.

Cited by 7 publications

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Largest known twin primes

Largest known twin primes

Estimating constellations among primes - I. Uniformity

Expected gaps between prime numbers

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