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Artykuły w czasopismach na temat „Maximal Gaps between Consecutive Primes”

Autor: Grafiati

Data publikacji: 3 czerwca 2025

Data aktualizacji: 31 lipca 2025

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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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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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6

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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11

Mullaji, Arun. "Can isolated removal of osteophytes achieve correction of varus deformity and gap-balance in computer-assisted total knee arthroplasty?" Bone & Joint Journal 102-B, no. 6_Supple_A (2020): 49–58. http://dx.doi.org/10.1302/0301-620x.102b6.bjj-2019-1597.r1.

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Aims The aims of this study were to determine the effect of osteophyte excision on deformity correction and soft tissue gap balance in varus knees undergoing computer-assisted total knee arthroplasty (TKA). Methods A total of 492 consecutive, cemented, cruciate-substituting TKAs performed for varus osteoarthritis were studied. After exposure and excision of both cruciates and menisci, it was noted from operative records the corrective interventions performed in each case. Knees in which no releases after the initial exposure, those which had only osteophyte excision, and those in which further

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12

Shaban, Aura Omondi. "Maximal Gaps between Consecutive Primes, the Number of Primes at a Given Magnitude, the Location of Nth Prime and the General Behavior of Primes." May 18, 2017. https://doi.org/10.5281/zenodo.581642.

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This paper is concerned with formulation and demonstration of new versions of equations that can help us resolve problems concerning maximal gaps between consecutive prime numbers, the number of prime numbers at a given magnitude and the location of nth prime number. There is also a mathematical argument on why prime numbers as elementary identities on their own respect behave the way they do. Given that the equations have already been formulated, there are worked out examples on numbers that represent different cohorts. This paper has therefore attempted to formulate an equation that approxim

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13

Sun, Yu-Chen, and Hao Pan. "On the gaps between consecutive primes." Forum Mathematicum, April 28, 2022. http://dx.doi.org/10.1515/forum-2021-0140.

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Abstract Let p n p_{n} denote the 𝑛-th prime. We prove that, for any m ≥ 1 m\geq 1 , there exist infinitely many 𝑛 such that p n - p n - m ≤ C m p_{n}-p_{n-m}\leq C_{m} for some large constant C m > 0 C_{m}>0 , and p n + 1 - p n ≥ c m ⁢ log ⁡ n ⁢ log ⁡ log ⁡ n ⁢ log ⁡ log ⁡ log ⁡ log ⁡ n log ⁡ log ⁡ log ⁡ n p_{n+1}-p_{n}\geq\frac{c_{m}\log n\log\log n\log\log\log\log n}{\log\log\log n} for some small constant c m > 0 c_{m}>0 . Furthermore, for any fixed positive integer 𝑙, there are many positive integers 𝑘 with ( k , l ) = 1 (k,l)=1 such that p ′ ⁢ ( k , l ) ≥ c ⁢ k ⋅ log ⁡ k ⁢ lo

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14

GOLDSTON, DANIEL ALAN, JANOS PINTZ, and CEM YALCIN YILDIRIM. "Positive proportion of small gaps between consecutive primes." Publicationes Mathematicae Debrecen, December 1, 2011, 433–44. http://dx.doi.org/10.5486/pmd.2011.5140.

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15

Cohen, Joel E. "Gaps Between Consecutive Primes and the Exponential Distribution." Experimental Mathematics, June 22, 2024, 1–10. http://dx.doi.org/10.1080/10586458.2024.2362348.

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16

Goldston, D. A., S. W. Graham, J. Pintz, and C. Y. Yildirim. "Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdos on Consecutive Integers." International Mathematics Research Notices, June 18, 2010. http://dx.doi.org/10.1093/imrn/rnq124.

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17

Goldston, Daniel A., Sidney W. Graham, Apoorva Panidapu, Janos Pintz, Jordan Schettler, and Cem Y. Yıldırım. "Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers II." Journal of Number Theory, July 2020. http://dx.doi.org/10.1016/j.jnt.2020.06.002.

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