Displaying similar documents to “On 5 -tuples of twin practical numbers”

The complex sum of digits function and primes

Jörg M. Thuswaldner (2000)

Journal de théorie des nombres de Bordeaux

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Canonical number systems in the ring of gaussian integers ℤ [ i ] are the natural generalization of ordinary q -adic number systems to ℤ [ i ] . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number b . In this paper we investigate the sum of digits function ν b of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the f -th power of a prime. Furthermore, we establish an Erdös-Kac type...

On the greatest prime factor of n 2 + 1

Jean-Marc Deshouillers, Henryk Iwaniec (1982)

Annales de l'institut Fourier

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There exist infinitely many integers n such that the greatest prime factor of n 2 + 1 is at least n 6 / 5 . The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.

P 2 in short intervals

Henryk Iwaniec, M. Laborde (1981)

Annales de l'institut Fourier

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For any sufficiently large real number x , the interval [ x , x + x 0 , 45 ] contains at least one integer having at most two prime factors .

Primitive divisors of Lucas and Lehmer sequences, II

Paul M. Voutier (1996)

Journal de théorie des nombres de Bordeaux

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Let α and β are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all α and β with h ( β / α ) ≤ 4 , the n -th element of these sequences has a primitive divisor for n > 30 . In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

On the almost Goldbach problem of Linnik

Jianya Liu, Ming-Chit Liu, Tianze Wang (1999)

Journal de théorie des nombres de Bordeaux

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Under the Generalized Riemann Hypothesis, it is proved that for any k ≥ 200 there is N k > 0 depending on k only such that every even integer ≥ N k is a sum of two odd primes and k powers of 2 .

Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett, Josef Blass, A. M. W. Glass, David B. Meronk, Ray P. Steiner (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let Λ = b 2 log α 2 - b 1 log α 1 - b 3 log α 3 ≠ 0 with b 1 , b 2 , b 3 positive integers and α 1 , α 2 , α 3 positive multiplicatively independent rational numbers greater than 1 . Let α j 1 = α j 1 / α j 2 with α j 1 , α j 2 coprime positive integers ( j = 1 , 2 , 3 ) . Let α j ≥ max { α j 1 , e } and assume that gcd ( b 1 , b 2 , b 3 ) = 1 . Let b ' = b 2 log α 1 + b 1 log α 2 b 2 log α 3 + b 3 log α 2 and assume that B ≥ max { 10 , log b ' } . We prove that either { b 1 , b 2 , b 3 } is c 4 , B -linearly dependent over ℤ (with respect to a 1 , a 2 , a 3 )...