## Finding prime pairs with particular gaps (2002)

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Venue: | Math. Comp |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Cutter02findingprime,

author = {Pamela A. Cutter},

title = {Finding prime pairs with particular gaps},

journal = {Math. Comp},

year = {2002},

volume = {70},

pages = {1737--1744}

}

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### Abstract

Abstract. By a prime gap of size g, we mean that there are primes p and p + g such that the g − 1 numbers between p and p + g are all composite. It is widely believed that infinitely many prime gaps of size g exist for all even integers g. However, it had not previously been known whether a prime gap of size 1000 existed. The objective of this article was to be the first to find a prime gap of size 1000, by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from 746 to 1000, and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size 1000 listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size 1000 with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form 6m +1, 12m − 1, 12m + 1 and their application

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Citation Context ...pair of primes with gap g, as desired. 2. Expected values Let πg(x) be the number of primes n up to x such that n + g is also prime. Then we have the following conjecture, due to Hardy and Littlewood =-=[3]-=-. Conjecture 1. If g is even, then πg(x) ∼ α � p − 1 p − 2 · x log 2 x where p|g p>2 α =2 � � 1 − 2 � p � �2 . p>2 1 − 1 p If we first just assume that the primality of the two numbers n, n + g near x... |

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Citation Context ... � n/5 for all n ≥ 2082. Granville and Ramaré have verified this theorem for 2082 ≤ n ≤ 10 10 by using a direct consequence of Kummer’s theorem, and for n ≥ 2 1617 by using bounds on exponential sums =-=[2]-=-. By using the following proposition of Granville and Ramaré, it becomes a practical computational problem to establish this theorem for 10 10 ≤ n ≤ 2 1617 . Proposition 1. If m is a positive integer ... |

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Citation Context ...than 1000 that were found. It then proceeds numerically, starting with gap 746. Gaps g listed with an asterisk are gaps that were already known to exist at the time these computations were performed (=-=[4]-=-, [7], [8]). These gaps were found again just to make the list complete. 4. Other calculations In addition to the prime gap problem, the primality testing ideas of Brillhart, Lehmer and Selfridge can ... |

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Citation Context ...1sjsg \Gamma 1, passes 2 pseudoprime tests; however, it is highly unlikely that there is such a problem with the data. Gaps g listed with an asterisk are gaps that are already known to exist ([2],[3],=-=[4]-=-), which I found again to make my list complete. 6 PAMELA CUTTER gap g a b # of values of i first several values of i 1000 360 227 2 360878, 472171 1002 360 227 3 548723, 948949, 969423 1100 396 250 2... |

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Citation Context ...H. Lehmer realized that if p \Gamma 1 = FR where F ? p 1=2 is factored, then there is a quick, practical way to show if p is prime. In 1975, this result was extended by Lehmer, Brillhart and Selfridge=-=[1]-=-, so that one only needs to have the factored part F ? p 1=3 to get a quick test. The test is based on the following theorem. Theorem 1. [BLS] Let N \Gamma 1 = FR, where F is factored, F ? N 1=3 and (... |

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Citation Context ...1000 that were found. It then proceeds numerically, starting with gap 746. Gaps g listed with an asterisk are gaps that were already known to exist at the time these computations were performed ([4], =-=[7]-=-, [8]). These gaps were found again just to make the list complete. 4. Other calculations In addition to the prime gap problem, the primality testing ideas of Brillhart, Lehmer and Selfridge can be ap... |

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(Show Context)
Citation Context ...928, D. H. Lehmer realized that if p − 1=FR where F>p 1/2 is factored, then there is a quick, practical way to show if p is prime. In 1975, this result was extended by Lehmer, Brillhart and Selfridge =-=[1]-=-, so that one only needs to have the factored part F>p 1/3 to get a quick test. The test is based on the following theorem. Theorem 1. Let N − 1=FR,whereF is factored, F>N 1/3 and (F +1) 2 �= N. Assum... |

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Citation Context ...that were found. It then proceeds numerically, starting with gap 746. Gaps g listed with an asterisk are gaps that were already known to exist at the time these computations were performed ([4], [7], =-=[8]-=-). These gaps were found again just to make the list complete. 4. Other calculations In addition to the prime gap problem, the primality testing ideas of Brillhart, Lehmer and Selfridge can be applied... |