1. bookVolume 114 (2017): Issue 5 (May 2017)
Journal Details
License
Format
Journal
eISSN
2353-737X
First Published
20 May 2020
Publication timeframe
1 time per year
Languages
English
access type Open Access

One Some Cancellation Algorithms II

Published Online: 26 May 2020
Volume & Issue: Volume 114 (2017) - Issue 5 (May 2017)
Page range: 97 - 103
Journal Details
License
Format
Journal
eISSN
2353-737X
First Published
20 May 2020
Publication timeframe
1 time per year
Languages
English
Abstract

We define b f(n) to be the smallest integer (a natural number) d such that numbers f(n1, n2, ..., nm), where n 1+n2+ ... + nmn are not divisible by d. For the given functions f : ℕm → ℕ, we will obtain the asymptotic characterisation of the sequence of the least non canceled numbers (bf (n))n­. In the case f :ℕ2 (k, l)→k3+l3∈ℕ, this characterisation can be rewritten in the terms of the permutations polynomials of finite commutative quotient ring ℤ/mℤ. There are situations in which we cannot expect formula for bf(n) to be simple, but we can provide the upper and lower bounds of it.

Keywords

[1] Arnold L.K., Benkoski S.J., McCabe B.J., The discriminator (a simple application of Bertrand’s postulate), Amer. Math. Monthly, 1985, 92, 275–277.10.1080/00029890.1985.11971598Search in Google Scholar

[2] Bremser P.S., Schumer P.D., Washington L.C., A note on the incongruence of consecutive integers to a fixed power, J. Number Theory, 1990, 35, No. 1, 105–108.10.1016/0022-314X(90)90106-2Search in Google Scholar

[3] Browkin J., Cao H-Q., Modifications of the Eratosthenes sieve, Colloq. Math. 135, 2014, 127–138.10.4064/cm135-1-10Search in Google Scholar

[4] Haque S., Shallit J., Discriminators and k-regular sequences, INTEGERS 16, 2016, Paper A76.Search in Google Scholar

[5] Moree P., Mullen G.L., Dickson polynomial discriminators, J. Number Theory 59, 1996, 88–105.10.1006/jnth.1996.0089Search in Google Scholar

[6] Moree P., Zumalacárregui A., Salajan’s conjecture on discriminating terms in an exponential sequence, J. Number Theory 160, 2016, 646–665.10.1016/j.jnt.2015.09.015Search in Google Scholar

[7] Sierpiński W., Elementary Theory of numbers, Ed. A. Schinzel, North-Holland 1988.Search in Google Scholar

[8] Zhi-Wei Sun, On funtions taking only prime values, J. Number Theory 133, 2013, 2794–2812.10.1016/j.jnt.2013.02.003Search in Google Scholar

[9] Tomski A., Zakarczemny M., On some cancellation algorithms, NNTDMM 23, 2017, 101–114.Search in Google Scholar

[10] Zieve M., A note on the discriminator, J. Number Theory 73, 1998, 122–138.10.1006/jnth.1998.2256Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo