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A higher rank Mersenne problem
 ANTS V Proceedings, Springer Lecture Notes in Computer Science
"... Abstract. The classical Mersenne problem has been a stimulating challenge to number theorists and computer scientists for many years. After briefly reviewing some of the natural settings in which this problem appears as a special case, we introduce an analogue of the Mersenne problem in higher rank, ..."
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Abstract. The classical Mersenne problem has been a stimulating challenge to number theorists and computer scientists for many years. After briefly reviewing some of the natural settings in which this problem appears as a special case, we introduce an analogue of the Mersenne problem in higher rank, in both a classical and an elliptic setting. Numerical evidence is presented for both cases, and some of the difficulties involved in developing even a heuristic understanding of the problem are discussed. 1.
DISTRIBUTED PRIMALITY PROVING AND THE PRIMALITY OF (2^3539+ 1)/3
, 1991
"... We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the fir ..."
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We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the first ordinary Titanic prime.
ZEROS OF 2ADIC LFUNCTIONS AND CONGRUENCES FOR CLASS NUMBERS AND FUNDAMENTAL UNITS
"... Abstract. We study the imaginary quadratic fields such that the Iwasawa λ2invariant equals 1, obtaining information on zeros of 2adic Lfunctions and relating this to congruences for fundamental units and class numbers. This paper explores the interplay between zeros of 2adic Lfunctions and cong ..."
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Abstract. We study the imaginary quadratic fields such that the Iwasawa λ2invariant equals 1, obtaining information on zeros of 2adic Lfunctions and relating this to congruences for fundamental units and class numbers. This paper explores the interplay between zeros of 2adic Lfunctions and congruences for fundamental units and class numbers of quadratic fields. An underlying motivation was to study the distribution of zeros of 2adic Lfunctions, the basic philosophy being that the location of the zeros causes restrictions on the 2adic behavior of the class numbers and fundamental units of real quadratic fields. Though the predicted restrictions involved the unit and class number together, numerical computations (we used PARI) revealed definite patterns for the unit and class number separately, which we were then able to prove. Several of these congruences are classical, but some of them seem to be new. We use the information obtained to study the distribution of the zeros, in particular their distances from 1 and 0. In a previous paper [14], one of us showed that, if (2 p +1)/3 is prime infinitely often, then it is possible to have zeros of 2adic Lfunctions
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"... For a fixed k ∈ N we consider a multiplicative basis in N such that every n ∈ N has the unique factorization as product of elements from the basis with the exponents not exceeding k. We introduce the notion of kmultiplicativity of arithmetic functions, and study several arithmetic functions natura ..."
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For a fixed k ∈ N we consider a multiplicative basis in N such that every n ∈ N has the unique factorization as product of elements from the basis with the exponents not exceeding k. We introduce the notion of kmultiplicativity of arithmetic functions, and study several arithmetic functions naturally defined in karithmetics. We study a generalized Euler function and prove analogs of the Wirsing and Delange theorems for karithmetics. MSC: Primary 11A51, 11A25; Secondary 11N37
Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.7 On a Compositeness Test for (2 p + 1)/3
"... In this note, we give a necessary condition for the primality of (2 p + 1)/3. 1 ..."
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In this note, we give a necessary condition for the primality of (2 p + 1)/3. 1