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On the Equation ax ≡ x (mod b)
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Jam Germain
Published/Copyright:
January 26, 2010
Abstract
Recently Jiménez-Urroz and Yebra [On the Equation ax ≡ x (mod bn)] constructed, for any given a and b, solutions x to the title equation. Moreover they showed how these can be lifted to higher powers of b to obtain a b-adic solution for certain integers b. In this paper we find all positive integer solutions x to the title equation, proving that, for given a and b, there are X/b + Ob(1) solutions x ≤ X. We also show how solutions may be lifted in more generality. Moreover we show that the construction of [Jiménez-Urroz and Yebra, On the Equation ax ≡ x (mod bn)] (and obvious modifications) cannot always find all solutions to ax ≡ x (mod b).
Keywords.: Congruence; solutions
Received: 2008-10-30
Revised: 2009-08-12
Accepted: 2009-08-22
Published Online: 2010-01-26
Published in Print: 2009-December
© de Gruyter 2009
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Articles in the same Issue
- On the Complexity of n-Player Hackenbush
- On the Equation ax ≡ x (mod b)
- A New Generalization of Fibonacci Sequence & Extended Binet's Formula
- On Rainbow Solutions to an Equation with a Quadratic Term
- 2-Pile Nim with a Restricted Number of Move-Size Imitations
- The Finite Heine Transformation and Conjugate Durfee Squares
- A Short Note on the Difference Between Inverses of Consecutive Integers Modulo p
- Factorization Results with Combinatorial Proofs
- Squares in (12 + m2) ⋯ (n2 + m2)
- On Monochromatic Sets of Integers Whose Diameters Form a Monotone Sequence
- Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins
- Une forme effective d'un théorème de Bateman sur la fonction phi d'Euler
- Differences of Multiple Fibonacci Numbers
