Simultaneous congruences

This paper considers the cyclic system of n ≥ 2 simultaneous congruences [Formula: see text] for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers qi ≥ 2, with gcd (q1q2 ⋯ qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807,…. If the positivity condition on the integers qi is dropped, then for r = 1 these systems of congruences, taken ( mod |qi|), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equations [Formula: see text] depending on three parameters (r, s, m).

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Searchable encryption schemes: With multiplication and simultaneous congruences

2012 9th International ISC Conference on Information Security and Cryptology ◽

10.1109/iscisc.2012.6408210 ◽

2012 ◽

Cited By ~ 1

Author(s):

Nol Premasathian ◽

Somsak Choto

Keyword(s):

Searchable Encryption ◽

Encryption Schemes ◽

Simultaneous Congruences

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The Chinese Remainder Theorem and Simultaneous Congruences

Number Theory in Science and Communication - Springer Series in Information Sciences ◽

10.1007/978-3-662-02395-2_16 ◽

1984 ◽

pp. 176-185

Author(s):

Manfred R. Schroeder

Keyword(s):

Chinese Remainder Theorem ◽

Simultaneous Congruences

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Congruences for Modular Forms mod 2 and QuaternionicS-ideal Classes

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-019-1 ◽

2018 ◽

Vol 70 (5) ◽

pp. 1076-1095 ◽

Cited By ~ 2

Author(s):

Kimball Martin

Keyword(s):

Modular Forms ◽

Hilbert Modular Forms ◽

Hilbert Modular ◽

Congruences For Modular Forms ◽

Simultaneous Congruences

AbstractWe prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin–Lehner eigenvalues. The proofs involve the notion of quaternionicS-ideal classes and the distribution of Atkin–Lehner signs among newforms.

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