<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>n</italic>, <italic>m</italic> be natural numbers with <italic>n</italic> ≥ 2. We say that an integer <italic>a</italic>, (<italic>a</italic>, <italic>n</italic>) = 1, is the <italic>m</italic>-th power residue modulo <italic>n</italic> if there exists an integer <italic>x</italic> such that <italic>xm</italic> ≡ <italic>a</italic>(mod <italic>n</italic>). Let <italic>C</italic>(<italic>n</italic>) denote the multiplicative group consisting of the residues modulo <italic>n</italic> which are relatively prime to <italic>n</italic>. Let <italic>s</italic>(<italic>n</italic>, <italic>m</italic>, <italic>a</italic>) be the smallest solution of the congruence <italic>xm</italic> ≡ <italic>a</italic>(mod <italic>n</italic>) in the set <italic>C</italic>(<italic>n</italic>). Let <italic>t</italic>(<italic>n</italic>, <italic>m</italic>, <italic>a</italic>) be the largest solution of the congruence x<italic>m</italic> ≡ <italic>a</italic>(mod <italic>n</italic>) in the set <italic>C</italic>(<italic>n</italic>). We will give an upper bound for <italic>s</italic>(<italic>n</italic>, <italic>m</italic>, <italic>a</italic>) and a lower bound for <italic>t</italic>(<italic>n</italic>, <italic>m</italic>, <italic>a</italic>).</abstract>

Let n, m be natural numbers with n ≥ 2. We say that an integer a, (a, n) = 1, is the m-th power residue modulo n if there exists an integer x such that xm ≡ a(mod n). Let C(n) denote the multiplicative group consisting of the residues modulo n which are relatively prime to n. Let s(n, m, a) be the smallest solution of the congruence xm ≡ a(mod n) in the set C(n). Let t(n, m, a) be the largest solution of the congruence xm ≡ a(mod n) in the set C(n). We will give an upper bound for s(n, m, a) and a lower bound for t(n, m, a).

Lower and upper bounds for solutions of the congruence x<italic>m</italic> ≡ <italic>a</italic>(mod <italic>n</italic>)

Institut of Matchematics, Faculty of Physis, Mathematics and Computer Science

Technical Transactions

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Lower and upper bounds for solutions of the congruence x ≡ (mod )

Let n, m be natural numbers with n ≥ 2. We say that an integer a, (a, n) = 1, is the m-th power residue modulo n if there exists an integer x such that xm ≡...

Lower and upper bounds for solutions of the congruence x^m ≡ a(mod n)

Published Online: May 29, 2020

Page range: 161 - 168

DOI: https://doi.org/10.4467/2353737XCT.17.194.7423

Keywords
smallest solution, largest solution, upper bound, lower bound, congruence relation, residue class, -th degree equation

© 2017 Maciej Zakarczemny, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Lower and upper bounds for solutions of the congruence xm ≡ a(mod n)

Published Online: May 29, 2020

Page range: 161 - 168

DOI: https://doi.org/10.4467/2353737XCT.17.194.7423

Keywordssmallest solution, largest solution, upper bound, lower bound, congruence relation, residue class, -th degree equation

© 2017 Maciej Zakarczemny, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Lower and upper bounds for solutions of the congruence x^m ≡ a(mod n)

Keywords
smallest solution, largest solution, upper bound, lower bound, congruence relation, residue class, -th degree equation