Primes that become composite after changing an arbitrary digit
Authors:
Michael Filaseta and Jeremiah Southwick
Journal:
Math. Comp. 90 (2021), 979-993
MSC (2020):
Primary 11Y11; Secondary 11A63, 11N36, 11P32, 11Y05
DOI:
https://doi.org/10.1090/mcom/3593
Published electronically:
November 24, 2020
MathSciNet review:
4194171
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Abstract | References | Similar Articles | Additional Information
Abstract: We show that a positive proportion of the primes have the property that if any one of its digits in base $10$, including its infinitely many leading $0$ digits, is replaced by a different digit, then the resulting number is composite.
- A. S. Bang, Taltheoretiske Undersøgelser, Tidsskrift for Mat., 4 (1886), 70–80, 130–137.
- John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr., Factorizations of $b^n \pm 1$, 2nd ed., Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, RI, 1988. $b=2,3,5,6,7,10,11,12$ up to high powers. MR 996414, DOI 10.1090/conm/022
- Peter Orno, James Propp, Alan Wayne, J. Phipps McGrath, Leon Gerber, Howard Eves, Murray S. Klamkin, Paul Erdos, H. Kestelman, Zane C. Motteler, Marlow Sholander, Richard Beigel, and Stanley J. Benkoski, Problems, Math. Mag. 52 (1979), no. 3, 179–184. MR 1572304
- Michael Filaseta, Mark Kozek, Charles Nicol, and John Selfridge, Composites that remain composite after changing a digit, J. Comb. Number Theory 2 (2010), no. 1, 25–36 (2011). MR 2895985
- H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
- Jackson Hopper and Paul Pollack, Digitally delicate primes, J. Number Theory 168 (2016), 247–256. MR 3515817, DOI 10.1016/j.jnt.2016.04.007
- Dan Ismailescu and Peter Seho Park, On pairwise intersections of the Fibonacci, Sierpiński, and Riesel sequences, J. Integer Seq. 16 (2013), no. 9, Article 13.9.8, 9. MR 3137937
- Murray S. Klamkin, G. Edgar, Richard A. Gibbs, R. P. Boas, Bernardo Recaman, Paul Erdos, Jeffrey Shallit, Danny Goldstein, Sidney Penner, St. Olaf Problem Group, W. Weston Meyer, L. Carlitz, and Richard Scoville, Problems, Math. Mag. 51 (1978), no. 1, 69–72. MR 1572252
- S. V. Konyagin, Numbers that become composite after changing one or two digits, Pioneer Jour. of Algebra, Number Theory and Appl. 6 (2013), 1–7.
- Neil J. A. Sloane, The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc. 65 (2018), no. 9, 1062–1074. MR 3822822
- Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)–University of South Carolina. MR 4144492
- Cameron L. Stewart, On divisors of Lucas and Lehmer numbers, Acta Math. 211 (2013), no. 2, 291–314. MR 3143892, DOI 10.1007/s11511-013-0105-y
- Terence Tao, A remark on primality testing and decimal expansions, J. Aust. Math. Soc. 91 (2011), no. 3, 405–413. MR 2900615, DOI 10.1017/S1446788712000043
- K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265–284 (German). MR 1546236, DOI 10.1007/BF01692444
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Additional Information
Michael Filaseta
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
MR Author ID:
66800
Email:
filaseta@math.sc.edu
Jeremiah Southwick
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
ORCID:
0000-0003-0751-8358
Email:
soutjt14@alumni.wfu.edu
Keywords:
Covering system,
digitally delicate primes,
density,
sieve
Received by editor(s):
January 13, 2020
Received by editor(s) in revised form:
January 30, 2020, and July 29, 2020
Published electronically:
November 24, 2020
Article copyright:
© Copyright 2020
American Mathematical Society