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On The Schinzel–Wójcik Problem Under Hypothesis H

Part of: Elementary number theory

Published online by Cambridge University Press:  03 November 2025

M. ANWAR
M. ANWAR*
Affiliation:
Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo 11566, Egypt. Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. e-mail: mohamedanwar@sci.asu.edu.eg
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Abstract

Given r non-zero rational numbers $a_1, \ldots, a_r$ which are not $\pm1$, we complete, under Hypothesis H, a characterisation of the Schinzel–Wójcik r-rational tuples (i.e. r-tuples of rational numbers for which the Schinzel–Wójcik problem has an affirmative answer) which satisfy that the sum of the exponents of the positive elements $a_i$ in the representation of $-1$ in terms of the elements $a_i$ in the multiplicative group $\langle a_1,\dots, a_r\rangle\subset \mathbb{Q}^*$ is even whenever $-1 \in \langle a_1,\dots, a_r\rangle.$

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 11A15: Power residues, reciprocity

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 13
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Cambridge Philosophical Society

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