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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 ..."
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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...
Determining Mill’s Constant and a Note on Honaker’s Problem
 Journal of Integer Sequences
, 2005
"... In 1947 Mills proved that there exists a constant A such that ⌊A3n ⌋ is a prime for every positive integer n. Determining A requires determining an effective Hoheisel type result on the primes in short intervals—though most books ignore this difficulty. Under the Riemann Hypothesis, we show that the ..."
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In 1947 Mills proved that there exists a constant A such that ⌊A3n ⌋ is a prime for every positive integer n. Determining A requires determining an effective Hoheisel type result on the primes in short intervals—though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of consecutive cubes and determine (given RH) that the least possible value of Mills’ constant A does begin with 1.3063778838. We calculate this value to 6850 decimal places by determining the associated primes to over 6000 digits and probable primes (PRPs) to over 60000 digits. We also apply the CramérGranville Conjecture to Honaker’s problem in a related context.
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 ..."
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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.
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
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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 ..."
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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.