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Journal articles on the topic 'Consecutive primes'

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Published: 2 June 2025

Last updated: 22 June 2025

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1

Peck, AS. "Differences Between Consecutive Primes." Proceedings of the London Mathematical Society 76, no. 1 (1998): 33–69. http://dx.doi.org/10.1112/s0024611598000021.

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2

Bazzanella, Danilo. "Primes Between Consecutive Powers." Rocky Mountain Journal of Mathematics 39, no. 2 (2009): 413–21. http://dx.doi.org/10.1216/rmj-2009-39-2-413.

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3

Banks, William D., Tristan Freiberg, and Caroline L. Turnage-Butterbaugh. "Consecutive primes in tuples." Acta Arithmetica 167, no. 3 (2015): 261–66. http://dx.doi.org/10.4064/aa167-3-4.

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4

Ruzsa, Imre Z. "Consecutive primes modulo 4." Indagationes Mathematicae 12, no. 4 (2001): 489–503. http://dx.doi.org/10.1016/s0019-3577(01)80038-0.

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5

Bazzanella, D. "Primes between consecutive squares." Archiv der Mathematik 75, no. 1 (2000): 29–34. http://dx.doi.org/10.1007/s000130050469.

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6

Radomskii, Artyom O. "Consecutive Primes in Short Intervals." Proceedings of the Steklov Institute of Mathematics 314, no. 1 (2021): 144–202. http://dx.doi.org/10.1134/s008154382104009x.

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7

Yu, Gang. "The Differences Between Consecutive Primes." Bulletin of the London Mathematical Society 28, no. 3 (1996): 242–48. http://dx.doi.org/10.1112/blms/28.3.242.

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8

Pan, Hao, and Zhi-Wei Sun. "Consecutive primes and Legendre symbols." Acta Arithmetica 190, no. 3 (2019): 209–20. http://dx.doi.org/10.4064/aa170810-29-11.

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9

Baker, R. C., and G. Harman. "The Difference Between Consecutive Primes." Proceedings of the London Mathematical Society s3-72, no. 2 (1996): 261–80. http://dx.doi.org/10.1112/plms/s3-72.2.261.

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10

Banks, William D., and Victor Z. Guo. "Consecutive primes and Beatty sequences." Journal of Number Theory 191 (October 2018): 158–74. http://dx.doi.org/10.1016/j.jnt.2018.04.003.

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11

Matomaki, K. "LARGE DIFFERENCES BETWEEN CONSECUTIVE PRIMES." Quarterly Journal of Mathematics 58, no. 4 (2007): 489–518. http://dx.doi.org/10.1093/qmath/ham021.

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12

Minggao, Lu. "The difference between consecutive primes." Acta Mathematica Sinica 1, no. 2 (1985): 109–18. http://dx.doi.org/10.1007/bf02560025.

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13

Mikawa, Hiroshi. "The differences between consecutive almost-primes." Tsukuba Journal of Mathematics 11, no. 2 (1987): 257–64. http://dx.doi.org/10.21099/tkbjm/1496160579.

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14

Baker, R. C., G. Harman, and J. Pintz. "The Difference Between Consecutive Primes, II." Proceedings of the London Mathematical Society 83, no. 3 (2001): 532–62. http://dx.doi.org/10.1112/plms/83.3.532.

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15

Dubner, H., T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann. "Ten consecutive primes in arithmetic progression." Mathematics of Computation 71, no. 239 (2001): 1323–28. http://dx.doi.org/10.1090/s0025-5718-01-01374-6.

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16

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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17

Mozzochi, C. J. "On the difference between consecutive primes." Journal of Number Theory 24, no. 2 (1986): 181–87. http://dx.doi.org/10.1016/0022-314x(86)90101-0.

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18

Dubner, Harvey, and Harry Nelson. "Seven consecutive primes in arithmetic progression." Mathematics of Computation 66, no. 220 (1997): 1743–50. http://dx.doi.org/10.1090/s0025-5718-97-00875-2.

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19

COPIL, VLAD, and LAURENŢIU PANAITOPOL. "ON THE RATIO OF CONSECUTIVE PRIMES." International Journal of Number Theory 06, no. 01 (2010): 203–10. http://dx.doi.org/10.1142/s1793042110002934.

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For n ≥ 1, let pn be the nth prime number and [Formula: see text] for n ≥ 1. Using several results of Erdős, we study the sequence (qn)n ≥ 1 and we prove similar results as for the sequence (dn)n ≥ 1, dn = pn+1 - pn. We also consider the sequence [Formula: see text] for n ≥ 1 and denote by Gn and An its geometrical and arithmetical averages. We prove that [Formula: see text] for n ≥ 4.

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20

Pollack, Paul, and Lola Thompson. "Arithmetic functions at consecutive shifted primes." International Journal of Number Theory 11, no. 05 (2015): 1477–98. http://dx.doi.org/10.1142/s1793042115400023.

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For each of the functions f ∈ {φ, σ, ω, τ} and every natural number K, we show that there are infinitely many solutions to the inequalities f(pn - 1) < f(pn+1 - 1) < ⋯ < f(pn+K - 1), and similarly for f(pn -1) > f(pn+1 - 1) > ⋯ > f(pn+K -1). We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

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21

Erdös, Paul, and Hans Riesel. "On admissible constellations of consecutive primes." BIT 28, no. 3 (1988): 391–96. http://dx.doi.org/10.1007/bf01941122.

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22

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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23

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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24

TANNER, NOAM. "STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS." International Journal of Number Theory 05, no. 01 (2009): 81–88. http://dx.doi.org/10.1142/s1793042109001918.

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In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring 𝔽q[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D.

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25

Cheng, Yuan-You Fu-Rui. "Explicit Estimate on Primes Between Consecutive Cubes." Rocky Mountain Journal of Mathematics 40, no. 1 (2010): 117–53. http://dx.doi.org/10.1216/rmj-2010-40-1-117.

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26

Peterson, I. "Progressing to a Set of Consecutive Primes." Science News 148, no. 11 (1995): 167. http://dx.doi.org/10.2307/3979508.

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27

Beslin, S. J., and E. V. Kortright. "Primes and consecutive sums in arithmetic progressions." International Journal of Computer Mathematics 49, no. 3-4 (1993): 157–62. http://dx.doi.org/10.1080/00207169308804226.

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28

Bazzanella, Danilo. "A Note on Primes Between Consecutive Powers." Rendiconti del Seminario Matematico della Università di Padova 121 (2009): 223–31. http://dx.doi.org/10.4171/rsmup/121-13.

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29

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 paper we extend a method of Erdős and Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of Hardy and Littlewood.

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30

Lemke Oliver, Robert J., and Kannan Soundararajan. "Unexpected biases in the distribution of consecutive primes." Proceedings of the National Academy of Sciences 113, no. 31 (2016): E4446—E4454. http://dx.doi.org/10.1073/pnas.1605366113.

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Although the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy−Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.

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31

Bazzanella, Danilo. "Some conditional results on primes between consecutive squares." Functiones et Approximatio Commentarii Mathematici 45, no. 2 (2011): 255–63. http://dx.doi.org/10.7169/facm/1323705816.

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32

Feng, Shaoji, and Xiaosheng Wu. "The k-tuple jumping champions among consecutive primes." Acta Arithmetica 156, no. 4 (2012): 325–39. http://dx.doi.org/10.4064/aa156-4-2.

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33

Odlyzko, Andrew M. "Iterated absolute values of differences of consecutive primes." Mathematics of Computation 61, no. 203 (1993): 373. http://dx.doi.org/10.1090/s0025-5718-1993-1182247-7.

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34

Bazzanella, Danilo. "Primes between consecutive squares and the Lindelöf hypothesis." Periodica Mathematica Hungarica 66, no. 1 (2012): 111–17. http://dx.doi.org/10.1007/s10998-013-1457-y.

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35

LAISHRAM, SHANTA, and T. N. SHOREY. "GRIMM'S CONJECTURE ON CONSECUTIVE INTEGERS." International Journal of Number Theory 02, no. 02 (2006): 207–11. http://dx.doi.org/10.1142/s1793042106000498.

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For positive integers n and k, it is possible to choose primes P1, P2,…, Pk such that Pi | (n + i) for 1 ≤ i ≤ k whenever n + 1, n + 2,…, n + k are all composites and n ≤ 1.9 × 1010. This provides a numerical verification of Grimm's Conjecture.

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36

THORNE, FRANK. "Bubbles of Congruent Primes." Mathematical Proceedings of the Cambridge Philosophical Society 157, no. 3 (2014): 443–56. http://dx.doi.org/10.1017/s0305004114000425.

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AbstractIn [15], Shiu proved that ifaandqare arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent toamoduloq. We generalize Shiu's theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent toamoduloq.

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37

Gnanam, A., M. A. Gopalan, and B. Anitha. "Sum of Squares of Consecutive Primes using Maximal Gap." International Journal of Mathematics Trends and Technology 19, no. 2 (2015): 108–11. http://dx.doi.org/10.14445/22315373/ijmtt-v19p514.

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38

Kumchev, A. "The difference between consecutive primes in an arithmetic progression." Quarterly Journal of Mathematics 53, no. 4 (2002): 479–501. http://dx.doi.org/10.1093/qjmath/53.4.479.

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39

Erdős, P. "Problems and results on the differences of consecutive primes." Publicationes Mathematicae Debrecen 1, no. 1 (2022): 33–37. http://dx.doi.org/10.5486/pmd.1949.1.1.06.

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40

Dudek, Adrian W., Loïc Grenié, and Giuseppe Molteni. "Primes in explicit short intervals on RH." International Journal of Number Theory 12, no. 05 (2016): 1391–407. http://dx.doi.org/10.1142/s1793042116500858.

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41

Koninck, Jean-Marie de, and Aleksandar Ivić. "On the Distance Between Consecutive Divisors of an Integer." Canadian Mathematical Bulletin 29, no. 2 (1986): 208–17. http://dx.doi.org/10.4153/cmb-1986-034-7.

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AbstractLet ω(n) denote the number of distinct prime divisors of a positive integer n. Then we define and where are primes and r ≥ 2. Similarly denote by the number of divisors of n and let be defined by where are the divisors of n. We prove that there exists constants A and B such that and

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42

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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43

Bazzanella, D. "The exceptional set for the distribution of primes between consecutive powers." Acta Mathematica Hungarica 116, no. 3 (2007): 197–207. http://dx.doi.org/10.1007/s10474-007-5174-y.

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44

Mikawa, Hiroshi. "On the intervals bewtween consecutive numbers that are sums of two primes." Tsukuba Journal of Mathematics 17, no. 2 (1993): 443–53. http://dx.doi.org/10.21099/tkbjm/1496162271.

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45

Diouf, Madieyna. "The gap gn between two consecutive primes satisfies gn= O(pn2/3)." International Journal of Contemporary Mathematical Sciences 15, no. 3 (2020): 113–20. http://dx.doi.org/10.12988/ijcms.2020.91439.

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46

Zhou, Nian Hong. "On the sum of the reciprocals of the differences between consecutive primes." Ramanujan Journal 47, no. 2 (2018): 427–33. http://dx.doi.org/10.1007/s11139-018-0034-7.

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47

Lü, Xiaodong, Zhiwei Wang, and Bin Chen. "On the smooth values of shifted almost-primes." International Journal of Number Theory 15, no. 01 (2019): 1–9. http://dx.doi.org/10.1142/s1793042118501683.

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Denote by [Formula: see text] (respectively, [Formula: see text]) the largest (respectively, the smallest) prime factor of the integer [Formula: see text]. In this paper, we prove a lower bound of almost-primes [Formula: see text] with [Formula: see text] such that [Formula: see text] for [Formula: see text]. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor.

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48

TSAI, MU-TSUN, and ALEXANDRU ZAHARESCU. "ON THE SUM OF CONSECUTIVE INTEGERS IN SEQUENCES II." International Journal of Number Theory 08, no. 05 (2012): 1281–99. http://dx.doi.org/10.1142/s1793042112500753.

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Let A be a sequence of natural numbers, rA(n) be the number of ways to represent n as a sum of consecutive elements in A, and MA(x) ≔ ∑n ≤ x rA(n). We give a new short proof of LeVeque's formula regarding MA(x) when A is an arithmetic progression, and then extend the proof to give asymptotic formulas for the case when A behaves almost like an arithmetic progression, and also when A is the set of primes in an arithmetic progression.

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49

Sujata S. Patil. "A Part of Oppermann’s Conjecture, Legendre’s Conjecture and Andrica’s Conjecture." International Journal of Scientific Research in Science and Technology 12, no. 2 (2025): 231–33. https://doi.org/10.32628/ijsrst25121200.

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In this paper we discuss a part of Oppermann’s Conjecture ”there is at least two primes between n2 − n to n2 and at least another two primes between n^2 to n^2+ n for n≥3.5×〖10〗^6 ”. A part of Legendre’s Conjecture ”there is at least two primes between n^2 to 〖(n+1)〗^2 for n ≥3.5×〖10〗^6 ” and a part of Andrica’s Conjecture states that ” √(p_(n+1) )− √(p_n ) < 1 for every pair of consecutive prime numbers p_n and p_(n+1) (of course, p_n< p_(n+1) ) for n≥3.5×〖10〗^6”. We propose a conjecture regarding the distribution of prime numbers.

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50

TSAI, MU-TSUN, and ALEXANDRU ZAHARESCU. "ON THE SUM OF CONSECUTIVE INTEGERS IN SEQUENCES." International Journal of Number Theory 08, no. 03 (2012): 643–52. http://dx.doi.org/10.1142/s1793042112500364.

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Let A be a set of natural numbers, rA(n) be the number of ways to represent n as a sum of consecutive elements in A, and MA(x) := ∑n ≤ x rA(n). Under various circumstances, we show that Mℕ(x) ~ ∑ MAi(x), where ℕ = ⊔ Ai is a partition of ℕ. In particular, we prove that the asymptotic formula holds when the Ai's are chosen such that Ai = {n ∈ ℕ : f(n) = i}, where f(n) can be one of the following: (1) f(n) = Ω(n), (2) f(n) = d(n), (3) f(n) = rℕ(n), (4) (assuming the Riemann Hypothesis) f(n) = ‖n‖ := min {∣n - p∣ : p ∈ ℙ} (the distance to the set of primes).

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