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Displaying similar documents to “On the almost Goldbach problem of Linnik”

On the fractional parts of x / n and related sequences. II

Bahman Saffari, R. C. Vaughan (1977)

Annales de l'institut Fourier

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As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of x h ( x ) where h is an arithmetical function (namely h ( n ) = 1 / n , h ( n ) = log n , h ( n ) = 1 / log n ) and n is an integer (or a prime order) running over the interval [ y ( x ) , x ) ] . The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

On the largest prime factor of n ! + 2 n - 1

Florian Luca, Igor E. Shparlinski (2005)

Journal de Théorie des Nombres de Bordeaux

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For an integer n ≥ 2 we denote by P ( n ) the largest prime factor of n . We obtain several upper bounds on the number of solutions of congruences of the form n ! + 2 n - 1 ≡ 0 ( mod q ) and use these bounds to show that lim sup n → ∞ P ( n ! + 2 n - 1 ) / n ≥ ( 2 π 2 + 3 ) / 18 .

Goldbach numbers in sparse sequences

Jörg Brüdern, Alberto Perelli (1998)

Annales de l'institut Fourier

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We show that for almost all n ∈ N , the inequality | p 1 + p 2 - exp ( ( log n ) γ ) | < 1 has solutions with odd prime numbers p 1 and p 2 , provided 1 < γ < 3 2 . Moreover, we give a rather sharp bound for the exceptional set. This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.