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On the fixed points of the map x↦xx modulo a prime, II
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)In: Finite Fields and Their Applications, ISSN 1071-5797, E-ISSN 1090-2465, Vol. 48, p. 141-159Article in journal (Refereed) Published
Abstract [en]

We study number theoretic properties of the map x↦xx(modp), where x∈{1,2,…,p−1}, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes p<N for which the map only has the trivial fixed point x=1. A key technical result, possibly of independent interest, is the existence of subsets Nq⊂{2,3,…,q−1} such that almost all k-tuples of distinct integers n1,n2,…,nk∈Nq are multiplicatively independent (if k is not too large), and |Nq|=q⋅(1+o(1)) as q→∞. For q a large prime, this is used to show that the number of solutions to a certain large and sparse system of Fq-linear forms {Ln}n=2 q−1 “behaves randomly” in the sense that |{v∈Fq d:Ln(v)=1,n=2,3,…,q−1}|∼qd(1−1/q)q∼qd/e. (Here d=π(q−1) and the coefficients of Ln are given by the exponents in the prime power factorisation of n.)

Place, publisher, year, edition, pages
Academic Press, 2017. Vol. 48, p. 141-159
Keyword [en]
Fixed points, Self-power maps
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-212204DOI: 10.1016/j.ffa.2017.07.011ISI: 000412613100011Scopus ID: 2-s2.0-85026918636OAI: oai:DiVA.org:kth-212204DiVA, id: diva2:1134365
Funder
Swedish Research Council, 621-2011-5498Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Note

QC 20170818

Available from: 2017-08-18 Created: 2017-08-18 Last updated: 2017-10-31Bibliographically approved

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