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Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
The Lattice Points of an NDimensional Tetrahedron
"... this paper we consider the set of integer lattice points inside or on the boundary of the ndimensional tetrahedron bounded by the hyperplanes (1:1) X 1 = 0; X 2 = 0; : : : ; Xn = 0 and (1:2) w 1 X 1 + w 2 Xn + : : : + wnXn = X where w 1 ; w 2 ; : : : ; wn are given positive real numbers. In other ..."
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this paper we consider the set of integer lattice points inside or on the boundary of the ndimensional tetrahedron bounded by the hyperplanes (1:1) X 1 = 0; X 2 = 0; : : : ; Xn = 0 and (1:2) w 1 X 1 + w 2 Xn + : : : + wnXn = X where w 1 ; w 2 ; : : : ; wn are given positive real numbers. In other words, the ntuples of nonnegative integers (a 1 ; a 2 ; : : : ; an ) for which (1:3) a 1 w 1 + a 2 w 2 + : : : + anwn X: Our interest in this question comes from number theory for, if each w j = log p j for some prime p j (with the p j 's distinct), then this is equivalent to considering the set of positive integers e
On the Divisibility of Fermat Quotients
"... We show that for a prime p the smallest a with a p−1 ̸ ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1). Keywords: sieve. Fermat ..."
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We show that for a prime p the smallest a with a p−1 ̸ ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1). Keywords: sieve. Fermat quotients, smooth numbers, Heilbronn sums, large AMS Mathematics Subject Classification: 11A07, 11L40, 11N25 1