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On a Question of Buium

Published online by Cambridge University Press:  20 November 2018

José Felipe Voloch
José Felipe Voloch*
Affiliation:
Department of Mathematics University of Texas Austin, Texas 78712 U.S.A., e-mail: voloch@math.utexas.edu
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Abstract

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We prove that ${{\left\{ \left( {{n}^{p}}-n \right)/P \right\}}_{p}}\in {{\Pi }_{p}}{{\text{F}}_{p}}$, with $p$ ranging over all primes, is independent of 1 over the integers, assuming a conjecture in elementary number theory generalizing the infinitude of Mersenne primes. This answers a question of Buium. We also prove a generalization.

Keywords

11A07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 2 , 01 June 2000 , pp. 236 - 238
Copyright
Copyright © Canadian Mathematical Society 2000

References

[B1] Buium, A., Geometry of p-jets. DukeMath. J. 82(1996), 349–357.Google Scholar
[B2] Buium, A., Arithmetic analogues of derivations. Algebra, J. 198(1997), 290–299.Google Scholar
[I] Ihara, Y., On Fermat quotient and “differentiation of numbers”. RIMS Kokyuroku 810(1992) 324–341, In Japanese; English translation by Hahn, S., Univ. of Georgia, (preprint).Google Scholar
[J] Johnson, W., On the nonvanishing of Fermat quotients (mod p). Reine, J. Angew.Math. 292(1977), 196–200.Google Scholar
[Sm] Smirnov, A. L., Hurwitz inequalities for number fields. St. PetersburgMath. J. 4(1993), 357–375.Google Scholar