<abstract xmlns="http://www.w3.org/1999/xhtml">We define <italic>b f</italic>(<italic>n</italic>) to be the smallest integer (a natural number) <italic>d</italic> such that numbers <italic>f</italic>(<italic>n1, n2, ..., nm</italic>), where <italic>n 1</italic>+<italic>n2</italic>+ ... + <italic>nm</italic> ≤ <italic>n</italic> are not divisible by <italic>d</italic>. For the given functions <italic>f</italic> : ℕ<italic>m</italic> → ℕ, we will obtain the asymptotic characterisation of the sequence of the least non canceled numbers (<italic>bf</italic> (<italic>n</italic>))<italic>n</italic>­ℕ. In the case <italic>f</italic> :ℕ<italic>2</italic> (<italic>k, l</italic>)→<italic>k3</italic>+<italic>l3</italic>∈ℕ, this characterisation can be rewritten in the terms of the permutations polynomials of finite commutative quotient ring ℤ/<italic>m</italic>ℤ. There are situations in which we cannot expect formula for <italic>bf</italic>(<italic>n</italic>) to be simple, but we can provide the upper and lower bounds of it.</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+ ... + nm ≤ n 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))n­ℕ. 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.

One Some Cancellation Algorithms II

Institute of Mathematics, Faculty of Physics, Mathematics and Computer Science

Technical Transactions

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{"article-title":"One Some Cancellation Algorithms II"}

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+ ... + nm ≤ n are not divisible by d. For...

One Some Cancellation Algorithms II

Published Online: May 26, 2020

Page range: 97 - 103

DOI: https://doi.org/10.4467/2353737XCT.17.073.6430

Keywords
cancellation algorithms, primes in arithmetic progression, quadratic and cubic forms

© 2020 Maciej Zakarczemny, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

One Some Cancellation Algorithms II

Published Online: May 26, 2020

Page range: 97 - 103

DOI: https://doi.org/10.4467/2353737XCT.17.073.6430

Keywordscancellation algorithms, primes in arithmetic progression, quadratic and cubic forms

© 2020 Maciej Zakarczemny, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 Public License.

Keywords
cancellation algorithms, primes in arithmetic progression, quadratic and cubic forms