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Finding prime pairs with particular gaps
 Math. Comp
, 2002
"... 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 ga ..."
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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
Iterated absolute values of differences of consecutive primes
 Mathematics of Computation
, 1993
"... Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that v ..."
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Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that verified this conjecture for k < tt(1013) » 3 x 10 ". It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences. 1.
Checking the Goldbach conjecture up to 4 × 1011
 MR 94a:11157, Zbl 783.11037
, 1993
"... Abstract. One of the most studied problems in additive number theory, Goldbach's conjecture, states that every even integer greater than or equal to 4 can be expressed as a sum of two primes. In this paper checking of this conjecture up to 4 • 10 " by the IBM 3083 mainframe with vector pro ..."
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Abstract. One of the most studied problems in additive number theory, Goldbach's conjecture, states that every even integer greater than or equal to 4 can be expressed as a sum of two primes. In this paper checking of this conjecture up to 4 • 10 " by the IBM 3083 mainframe with vector processor is reported. 1.
Journal of Integer Sequences, Vol. 6 (2003), Article 03.3.1
"... The interval from 10 15 to 5 × 10 16 was searched for first occurrence prime gaps and maximal prime gaps. One hundred and twentytwo new first occurrences were found, including four new maximal gaps, leaving 1048 as the smallest gap whose first occurrence remains uncertain. The first occurrence of a ..."
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The interval from 10 15 to 5 × 10 16 was searched for first occurrence prime gaps and maximal prime gaps. One hundred and twentytwo new first occurrences were found, including four new maximal gaps, leaving 1048 as the smallest gap whose first occurrence remains uncertain. The first occurrence of any prime gap of 1000 or greater was found to be the maximal gap of 1132 following the prime 1693182318746371. A maximal gap of 1184 follows the prime 43841547845541059. More extensive tables of prime gaps are maintained at
Article electronically published on February 13, 1999 NEW MAXIMAL PRIME GAPS AND FIRST OCCURRENCES
"... Abstract. The search for first occurrences of prime gaps and maximal prime gaps is extended to 10 15. New maximal prime gaps of 806 and 906 are found, and sixtytwo previously unpublished first occurrences are found for gaps varying from 676 to 906. 1. ..."
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Abstract. The search for first occurrences of prime gaps and maximal prime gaps is extended to 10 15. New maximal prime gaps of 806 and 906 are found, and sixtytwo previously unpublished first occurrences are found for gaps varying from 676 to 906. 1.
New Prime Gaps Between 10^15 and 5 × 10^16
, 2003
"... The interval from 10 to 5 10 was searched for rst occurrence prime gaps and maximal prime gaps. One hundred and twentytwo new rst occurrences were found, including four new maximal gaps, leaving 1048 as the smallest gap whose rst occurrence remains uncertain. The rst occurrence of any pr ..."
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The interval from 10 to 5 10 was searched for rst occurrence prime gaps and maximal prime gaps. One hundred and twentytwo new rst occurrences were found, including four new maximal gaps, leaving 1048 as the smallest gap whose rst occurrence remains uncertain. The rst occurrence of any prime gap of 1000 or greater was found to be the maximal gap of 1132 following the prime 1693182318746371. A maximal gap of 1184 follows the prime 43841547845541059. More extensive tables of prime gaps are maintained at http://www.trnicely.net.
Short effective intervals containing primes
, 2000
"... We prove that every interval xð1 D1Þ; x contains a prime number with D 28 314 000 and provided xX10 726 905 041: The proof combines analytical, sieve and algorithmical methods. ..."
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We prove that every interval xð1 D1Þ; x contains a prime number with D 28 314 000 and provided xX10 726 905 041: The proof combines analytical, sieve and algorithmical methods.