Results 1 
9 of
9
Almost All Primes Can be Quickly Certified
"... This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], SolovayStrassen [SSI, and Rabin [R] ..."
Abstract

Cited by 69 (4 self)
 Add to MetaCart
This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], SolovayStrassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption. Thc test terminates in expected polynomial time on all but at most an exponentially vanishing fraction of the inputs of length k, for every k. This result implies: • There exist an infinite set of primes which can be recognized in expected polynomial time. • Large certified primes can be generated in expected polynomial time. Under a very plausible condition on the distribution of primes in "small" intervals, the proposed algorithm can be shown'to run in expected polynomial time on every input. This
Chains of large gaps between consecutive primes
 Adv. in Math
, 1981
"... ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log log log 10gx(loglog logx)2, where c = eY and y is Euler's constant. Here, this result is shown with ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log log log 10gx(loglog logx)2, where c = eY and y is Euler's constant. Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4 / Co e4/co = 3. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus. 1.
Some Conjectures on the Gaps between Consecutive Primes
 in the Distribution of Prime Numbers ", in Proc. of the 8th Joint EPSAPS Int.Conf. Physics
"... Five conjectures on the gaps between consecutive primes are formulated. One expresses the number of twins below a given bound directly by ß(N ). These conjectures are compared with the computer results and a good agreement is found. 1. Introduction. In 1922 Hardy and Littlewood [1] have proposed ab ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
Five conjectures on the gaps between consecutive primes are formulated. One expresses the number of twins below a given bound directly by ß(N ). These conjectures are compared with the computer results and a good agreement is found. 1. Introduction. In 1922 Hardy and Littlewood [1] have proposed about 15 conjectures. The conjecture B of their paper states: There are infinitely many primes pairs (p; p 0 ), where p 0 = p + d, for every even d. If ß d (N) denotes the number of pairs less than N , then ß d (N) ¸ 2c 2 N log 2 (N) Y pjd p \Gamma 1 p \Gamma 2 : (1) Here the constant c 2 is defined in the following way 1 : c 2 j Y p?2 ` 1 \Gamma 1 (p \Gamma 1) 2 ' = 0:66016 : : : (2) The computer results of the search for pairs of primes separated by a distance d and smaller than N for N = 2 22 ; 2 24 ; : : : ; 2 40 ß 1:1 \Theta 10 12 are shown in the Fig.1. The characteristic oscillating pattern of points is caused by the product J(d) = Y pjd;p?2 p \Gamm...
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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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 verified this ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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.
GAPS BETWEEN INTEGERS WITH THE SAME PRIME FACTORS
"... Abstract. We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers a
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers a<cis ≫ a 1/2−ɛ, and it is shown that this lower bound is best possible. Dressler’s conjecture is verified for values of a and c up to 7 · 10 13. 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 ..."
Abstract
 Add to MetaCart
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.
Maximal Gaps Between Prime kTuples: A Statistical Approach
"... Combining the HardyLittlewood ktuple conjecture with a heuristic application of extremevalue statistics, we propose a family of estimator formulas for predicting maximalgapsbetweenprimektuples. Extensivecomputationsshowthattheestimator alog(x/a)−ba satisfactorily predicts the maximal gaps below ..."
Abstract
 Add to MetaCart
Combining the HardyLittlewood ktuple conjecture with a heuristic application of extremevalue statistics, we propose a family of estimator formulas for predicting maximalgapsbetweenprimektuples. Extensivecomputationsshowthattheestimator alog(x/a)−ba satisfactorily predicts the maximal gaps below x, in most cases within an error of ±2a, where a = Cklog k x is the expected average gap between the same type of ktuples. Heuristics suggest that maximal gaps between prime ktuples near x are asymptotically equal to alog(x/a), and thus have the order O(log k+1 x). The distributionofmaximalgapsaroundthe“trend”curvealog(x/a)isclosetotheGumbel distribution. We explore two implications of this model of gaps: record gaps between primes and Legendretype conjectures for prime ktuples. 1