Skip to main content

Variants of Korselt’s Criterion

  • Thomas Wright (a1)

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer a, there are infinitely many n ∊ ℕ such that for each prime factor p | n, we have pa | na. This can be seen as a generalization of Carmichael numbers, which are integers n such that p − 1 | n − 1 for every p | n.

Hide All
[AGP] Alford, W. R., Granville, A., and Pomerance, C., There are infinitely many Carmichael numbers. Ann. of Math. (2) 139(1994), no. 3, 703722. http://dx.doi.Org/10.2307/2118576
[BP] Banks, W. D. and Pomerance, C., On Carmichael numbers in arithmetic progressions. J. Aust. Math. Soc. 88(2010), no. 3, 313321. http://dx.doi.Org/10.1017/S1446788710000169
[Ca] Carmichael, R. D., Note on a new number theory function. Bull. Amer. Math. Soc. 16(1910), no. 5, 232238. http://dx.doi.Org/10.1090/S0002-9904-1910-01892-9
[EK] Van Emde Boas, P. and Kruyswijk, D., A combinatorial problem on finite Abelian groups III. Zuivere Wisk. (1969) (Math. Centrum, Amsterdam).
[EPT] Ekstrom, A., Pomerance, C., and Thakur, D. S., Infinitude of elliptic Carmichael numbers. J. Aust. Math. Soc. 92(2012), no. 1, 4560. http://dx.doi.Org/10.1017/S1446788712000080
[HB] Heath-Brown, D. R., Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Philos. Soc. 83(1978), no. 3, 357375. http://dx.doi.Org/10.1017/S0305004100054657
[Ko] Korselt, A., Problème chinois. L'intermédinaire des mathématiciens 6(1899), 142143.
[Ma] Matomâki, K., Carmichael numbers in arithmetic progressions. J. Aust. Math. Soc. 94(2013), no. 2, 268275. http://dx.doi.Org/10.1017/S1446788712000547
[Me] Meshulam, R., An uncertainty inequality and zero subsums. Discrete Math. 84(1990), no. 2, 197200. http://dx.doi.Org/10.1016/0012-365X(90)90375-R
[MV] Montgomery, H. and Vaughan, R., Multiplicative number theory I: Classical theory. Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007.
[RS] Rosser, J. B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers. Illinois J. Math. 6(1962), 6494.
[Wr2] Rosser, J. B. and Schoenfeld, L., Infinitely many Carmichael numbers in arithmetic progressions. Bull. Lond. Math. Soc. 45(2013), no. 5, 943952. http://dx.doi.Org/10.1112/blms/bdtO13
[Xy] Xylouris, T., Ûber die Linniksche Konstante'. arxiv:0906.2749
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed