93 research outputs found
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
On -coordinates of Pell equations which are repdigits
Let be a given integer. In this paper, we show that there only
finitely many positive integers which are not squares, such that the Pell
equation has two positive integer solutions with the
property that their -coordinates are base -repdigits. Recall that a base
-repdigit is a positive integer all whose digits have the same value when
written in base . We also give an upper bound on the largest such in
terms of .Comment: To appear in The Fibonacci Quarterly Journa
On the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n
In this paper, we give all the solutions of the Diophantine equation
x^2+7^{alpha}.11^{beta}=y^n, in nonnegative integers x, y, n>=3 with x and y
coprime, except for the case when alpha.x is odd and beta is even.Comment: to appear in Miskolc Mathematical Notes; MSC key words and phrases:
Exponential equations, Primitive divisors of Lucas sequence
Grained integers and applications to cryptography
To meet the requirements of the modern communication society, cryptographic techniques are of central importance. In modern cryptography, we try to build cryptographic primitives, whose security can be reduced to solving a particular number theoretic problem for which no fast algorithmic method is known by now. Thus, any advance in the understanding of the nature of such problems indirectly gives insight in the analysis of some of the most practical cryptographic techniques. In this work we analyze exactly this aspect much more deeply: How can we use some of the purely theoretical results in number theory to answer very practical questions on the security of widely used cryptographic algorithms and how can we use such results in concrete implementations? While trying to answer these kinds of security-related questions, we always think two-fold: From a cryptographic, security-ensuring perspective and from a cryptanalytic one. After we outlined -- with a special focus on the historical development of these results -- the necessary analytic and algorithmic foundations of number theory, we first delve into the question how point addition on certain elliptic curves can be done efficiently. The resulting formulas have their application in the cryptanalysis of crypto systems that are insecure if factoring integers can be done efficiently. The rest of the thesis is devoted to the study of integers, all of whose prime factors are neither too small nor too large. We show with the help of two applications how one can use the properties of such kinds of integers to answer very practical questions in the design and the analysis of cryptographic primitives: The optimization of a hardware-realization of the cofactorization step of the General Number Field Sieve and the analysis of different standardized key-generation algorithms
Primitive divisors on twists of the Fermat cubic
We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3+v3=m, with m cube-free, all the terms beyond the first have a primive divisor
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