1. bookVolume 115 (2018): Issue 7 (July 2018)
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

A Note On Browkin’s and Cao’s Cancellation Algorithm

Published Online: 21 May 2020
Volume & Issue: Volume 115 (2018) - Issue 7 (July 2018)
Page range: 153 - 165
Received: 24 Apr 2018
Journal Details
License
Format
Journal
eISSN
2353-737X
First Published
20 May 2020
Publication timeframe
1 time per year
Languages
English
Abstract

In this paper, we follow our generalisation of the cancellation algorithm described in our previous paper [A. Tomski, M. Zakarczemny, On some cancellation algorithms, NNTDM. 23, 2017, p. 101–114]. for f being a natural-valued function defined on ℕs s, ≥1 we remove the divisors of all possible values of f in the points in which the sum of coordinates is less than or equal to n. The least non-cancelled number is called the discriminator Df(n). We find formulas, or at least an estimation for this discriminator, in the case of a broad class of sequences.

Keywords

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

[2] Bremser P. S., Schumer P.D., Washington L.C., A note on the incon-gruence of consecutive integers to a fixed power, J. Number Theory 35, no. 1, 1990, 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., The incongruence of consecutive values of polynomials, Finite Fields Appl., 2, 1996, 321–335.10.1006/ffta.1996.0020Search in Google Scholar

[7] Moree P., Zumalacarregui A., Sălăjan’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

[8] Sierpiński W., Elementary theory of numbers, Warszawa 1964.Search in Google Scholar

[9] Sierpiński W., Sur certaines hypothèses concernant les nombres premiers, Acta Arith., 4, no. 1, 1958, 20–30.10.4064/aa-4-1-20-30Search in Google Scholar

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

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

[12] Zakarczemny M., On some cancellation algorithms, II, Technical Transactions, vol. 5/2017.10.4467/2353737XCT.17.073.6430Search in Google Scholar

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

[14] Sloane N.J., The On-Line Encyclopedia of Integer Sequences, available online at https://oeis.org (access: 05.10.2018).Search in Google Scholar

[15] Ciolan A., Moree P., Browkin’s Discriminator Conjecture, 2017, available online at https://arxiv.org/abs/1707.02183 (access: 05.10.2018).Search in Google Scholar

[16] Melfi G., On the conditional infiniteness of primitive weird numbers, J. Number Theory, 147, 2015, 508–514.10.1016/j.jnt.2014.07.024Search in Google Scholar

[17] https://webspace.ship.edu/msrenault/fibonacci/fib.htm (access: 05.10.2018).Search in Google Scholar

[18] https://www.mathpages.com/home/kmath078/kmath078.htm (access: 05.10.2018).Search in Google Scholar

[19] Baker R.C., Harman G., Pintz J., The difference between consecutive primes, II, Proc. Lond. Math Soc., 83, 2001, 532–562.10.1112/plms/83.3.532Search in Google Scholar

[20] Oppermann L., Om vor Kundskab om Primtallenes Moengde mellem givne Groenser, Overs. Dansk. Vidensk. Selsk. Forh., 1882, 169–179.Search in Google Scholar

[21] Cramer H., On the order of the magnitude of the difference between consecutive prime number, Acta Arith. 2, 1936, 23–46.10.4064/aa-2-1-23-46Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo