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Relevant bibliographies by topics / Maximal Gaps between Consecutive Primes

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Academic literature on the topic 'Maximal Gaps between Consecutive Primes'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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Journal articles on the topic "Maximal Gaps between Consecutive Primes"

1

Kourbatov, Alexei, and Marek Wolf. "Predicting Maximal Gaps in Sets of Primes." Mathematics 7, no. 5 (2019): 400. http://dx.doi.org/10.3390/math7050400.

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Let q > r ≥ 1 be coprime integers. Let P c = P c ( q , r , H ) be an increasing sequence of primes p satisfying two conditions: (i) p ≡ r (mod q) and (ii) p starts a prime k-tuple with a given pattern H. Let π c ( x ) be the number of primes in P c not exceeding x. We heuristically derive formulas predicting the growth trend of the maximal gap G c ( x ) = max p ′ ≤ x ( p ′ − p ) between successive primes p , p ′ ∈ P c. Extensive computations for primes up to 10 14 show that a simple trend formula G c ( x ) ∼ x π c ( x ) · ( log π c ( x ) + O k ( 1 ) ) works well for maximal gaps between ini

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2

Maier, Helmut, and Carl Pomerance. "Unusually large gaps between consecutive primes." Transactions of the American Mathematical Society 322, no. 1 (1990): 201–37. http://dx.doi.org/10.1090/s0002-9947-1990-0972703-x.

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3

Pintz, János. "Very Large Gaps between Consecutive Primes." Journal of Number Theory 63, no. 2 (1997): 286–301. http://dx.doi.org/10.1006/jnth.1997.2081.

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4

GOLDSTON, D. A., and A. H. LEDOAN. "JUMPING CHAMPIONS AND GAPS BETWEEN CONSECUTIVE PRIMES." International Journal of Number Theory 07, no. 06 (2011): 1413–21. http://dx.doi.org/10.1142/s179304211100471x.

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The most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given x. In 1999 Odlyzko, Rubinstein and Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,…. As a step toward proving this conjecture they introduced a second weaker conjecture that any fixed prime p divides all sufficiently large jumping champions. In this

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5

Goldston, Daniel Alan, János Pintz, and Cem Yalçın Yıldırım. "Primes in tuples IV: Density of small gaps between consecutive primes." Acta Arithmetica 160, no. 1 (2013): 37–53. http://dx.doi.org/10.4064/aa160-1-3.

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Aaccagnini, Alessandro. "A note on large gaps between consecutive primes in arithmetic progressions." Journal of Number Theory 42, no. 1 (1992): 100–102. http://dx.doi.org/10.1016/0022-314x(92)90111-2.

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7

Pintz, J. "On a Conjecture of Erdős, Pólya and Turán on Consecutive Gaps Between Primes." Analysis Mathematica 44, no. 2 (2018): 263–71. http://dx.doi.org/10.1007/s10476-018-0210-4.

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8

Brown, Steven. "Distance between consecutive elements of the multiplicative group of integers modulo n." Notes on Number Theory and Discrete Mathematics 30, no. 1 (2024): 81–99. http://dx.doi.org/10.7546/nntdm.2024.30.1.81-99.

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For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\ZZ{P}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a circle of perimeter $P$. These points considered with wrap around modulo $P$ are those not marked by the Eratosthenes sieve algorithm applied to all primes less than or equal to $p$. In this paper, we are mostly concerned with providing formulas to count the number of gaps of a given even length $D$ in $U(P)$ which we note $K(D,P)$. This work, presented with different notations

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9

Mayer, B. L., and L. H. A. Monteiro. "A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms." Complexity 2020 (September 10, 2020): 1–5. http://dx.doi.org/10.1155/2020/1480890.

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Let a d -prime be a positive integer number with d divisors. From this definition, the usual prime numbers correspond to the particular case d = 2 . Here, the seemingly random sequence of gaps between consecutive d -primes is numerically investigated. First, the variability of the gap sequences for d ∈ 2,3 , … , 11 is evaluated by calculating the informational entropy. Then, these sequences are mapped into graphs by employing two visibility algorithms. Computer simulations reveal that the degree distribution of most of these graphs follows a power law. Conjectures on how some topological featu

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10

Fraile, Alberto, Roberto Martínez, and Daniel Fernández. "Jacob’s Ladder: Prime Numbers in 2D." Mathematical and Computational Applications 25, no. 1 (2020): 5. http://dx.doi.org/10.3390/mca25010005.

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Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense variety of problems. In this work, we present a simple representation of prime numbers in two dimensions that allows us to formulate a number of conjectures that may lead to important avenues in the field of research on prime numbers. In particular, although the zeroes in our representation grow in a somewhat erratic, hardly predictable way, the gaps between them

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Book chapters on the topic "Maximal Gaps between Consecutive Primes"

1

Pintz, János. "On the Ratio of Consecutive Gaps Between Primes." In Analytic Number Theory. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-22240-0_17.

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2

"Unusually large gaps between consecutive primes." In Théorie des nombres / Number Theory. De Gruyter, 1989. http://dx.doi.org/10.1515/9783110852790.625.

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"Small and large gaps between consecutive primes." In The Life of Primes in 37 Episodes. American Mathematical Society, 2021. http://dx.doi.org/10.1090/mbk/139/17.

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