Geodesic
Prime avoiding numbers form a basis of order $2$
Sbornik. Mathematics, Tome 215 (2024) no. 5, pp. 612-633 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a positive integer $n$ we denote by $F(n)$ the distance of $n$ to the nearest prime number. Using the technique from the recent paper “Long gaps in sieved sets” by Ford, Konyagin, Maynard, Pomerance and Tao (J. Eur. Math. Soc., 23:2 (2021), 667–700) we prove that every sufficiently large positive integer $N$ can be represented as a sum $N=n_1+n_2$, where $F(n_i) \geq (\log N)(\log\log N)^{1/325565}$ for $i=1,2$. This improves the corresponding ‘trivial’ statement where only the inequality $F(n_i)\gg \log N$ is assumed. Bibliography: 17 titles.
Keywords: prime numbers, basis, sieving. 
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M. R. Gabdullin; A. O. Radomskii. Prime avoiding numbers form a basis of order $2$. Sbornik. Mathematics, Tome 215 (2024) no. 5, pp. 612-633. http://geodesic.mathdoc.fr/item/SM_2024_215_5_a1/

[1] R. A. Rankin, “The difference between consecutive prime numbers”, J. London Math. Soc., 13:4 (1938), 242–247 | DOI | MR | Zbl

[2] E. Westzynthius, “Über die Verteilung der Zahlen, die zu den $n$ ersten Primzahlen teilerfremd sind”, Comment. Phys.-Math. Soc. Sci. Fenn., 5:25 (1931), 1–37 | Zbl

[3] P. Erdős, “On the difference of consecutive primes”, Quart. J. Math. Oxford Ser., 6 (1935), 124–128 | DOI | Zbl

[4] J. Pintz, “Very large gaps between consecutive primes”, J. Number Theory, 63:2 (1997), 286–301 | DOI | MR | Zbl

[5] K. Ford. B. Green, S. Konyagin and T. Tao, “Large gaps between consecutive prime numbers”, Ann. of Math. (2), 183:3 (2016), 935–974 | DOI | MR | Zbl

[6] J. Maynard, “Large gaps between primes”, Ann. of Math. (2), 183:3 (2016), 915–933 | DOI | MR | Zbl

[7] P. Erdős, “Some of my favourite unsolved problems”, A tribute to Paul Erdős, Cambridge Univ. Press, Cambridge, 1990, 467–478 | DOI | MR | Zbl

[8] K. Ford, B. Green, S. Konyagin, J. Maynard and T. Tao, “Long gaps between primes”, J. Amer. Math. Soc., 31:1 (2018), 65–105 | DOI | MR | Zbl

[9] H. Cramér, “On the order of magnitude of the difference between consecutive prime numbers”, Acta Arith., 2 (1936), 23–46 | DOI | Zbl

[10] A. Granville, “Harald Cramér and the distribution of prime numbers”, Scand. Actuar. J., 1995:1 (1995), 12–28 | DOI | MR | Zbl

[11] W. Banks, K. Ford and T. Tao, “Large prime gaps and probabilistic models”, Invent. Math., 233:3 (2023), 1471–1518 | DOI | MR | Zbl

[12] R. C. Baker, G. Harman and J. Pintz, “The difference between consecutive primes. II”, Proc. London Math. Soc. (3), 83:3 (2001), 532–562 | DOI | MR | Zbl

[13] K. Ford, D. R. Heath-Brown and S. Konyagin, “Large gaps between consecutive prime numbers containing perfect powers”, Analytic number theory, Springer, Cham, 2015, 83–92 | DOI | MR | Zbl

[14] H. Maier and M. Th. Rassias, “Large gaps between consecutive prime numbers containing perfect $k$th powers of prime numbers”, J. Funct. Anal., 272:6 (2017), 2659–2696 | DOI | MR | Zbl

[15] H. Maier and M. Th. Rassias, “Prime avoidance property of $k$th powers of prime numbers with Beatty sequence”, Discrete mathematics and applications, Springer Optim. Appl., 165, Springer, Cham, 2020, 397–404 | DOI | MR | Zbl

[16] K. Ford, S. Konyagin, J. Maynard, C. B. Pomerance and T. Tao, “Long gaps in sieved sets”, J. Eur. Math. Soc. (JEMS), 23:2 (2021), 667–700 | DOI | MR | Zbl

[17] K. Ford, S. Konyagin, J. Maynard, C. B. Pomerance and T. Tao, “Corrigendum: Long gaps in sieved sets”, J. Eur. Math. Soc. (JEMS), 25:6 (2023), 2483–2485 | DOI | MR | Zbl