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On the diophantine equation ( x m + 1 ) ( x n + 1 ) = y ²

Maohua Le (1997)

Acta Arithmetica

1. Introduction. Let ℤ, ℕ, ℚ be the sets of integers, positive integers and rational numbers respectively. In [7], Ribenboim proved that the equation    (1) ( x m + 1 ) ( x n + 1 ) = y ² , x,y,m,n ∈ ℕ, x > 1, n > m ≥ 1, has no solution (x,y,m,n) with 2|x and (1) has only finitely many solutions (x,y,m,n) with 2∤x. Moreover, all solutions of (1) with 2∤x satisfy max(x,m,n) < C, where C is an effectively computable constant. In this paper we completely determine all solutions of (1) as follows.   Theorem. Equation (1)...

On the discrepancy of sequences associated with the sum-of-digits function

Gerhard Larcher, N. Kopecek, R. F. Tichy, G. Turnwald (1987)

Annales de l'institut Fourier

If w = ( q k ) k ∈ N denotes the sequence of best approximation denominators to a real α , and s α ( n ) denotes the sum of digits of n in the digit representation of n to base w , then for all x irrational, the sequence ( s α ( n ) · x ) n ∈ N is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if α has bounded continued fraction coefficients.

On the distance between generalized Fibonacci numbers

Jhon J. Bravo, Carlos A. Gómez, Florian Luca (2015)

Colloquium Mathematicae

For an integer k ≥ 2, let ( F ₙ ( k ) ) ₙ be the k-Fibonacci sequence which starts with 0,..., 0,1 (k terms) and each term afterwards is the sum of the k preceding terms. This paper completes a previous work of Marques (2014) which investigated the spacing between terms of distinct k-Fibonacci sequences.

On the distribution of p α modulo one

Xiaodong Cao, Wenguang Zhai (1999)

Journal de théorie des nombres de Bordeaux

In this paper, we give a new upper-bound for the discrepancy D ( N ) : = sup 0 ≤ γ ≤ 0 | ∑ p ≤ / N p α ≤ γ 1 - π ( N ) γ | for the sequence ( p α ) , when 5 / 3 ≤ α &gt; 3 and α ≠ 2 .

On the exponential local-global principle

Boris Bartolome, Yuri Bilu, Florian Luca (2013)

Acta Arithmetica

Skolem conjectured that the "power sum" A(n) = λ₁α₁ⁿ + ⋯ + λₘαₘⁿ satisfies a certain local-global principle. We prove this conjecture in the case when the multiplicative group generated by α₁,...,αₘ is of rank 1.

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

Bahman Saffari, R. C. Vaughan (1976)

Annales de l'institut Fourier

This paper and its sequels deal with a new concept of distributions modulo one which is connected with the Dirichlet divisor and similar problems. Each of the theorems has some independent interest, and in addition some of the techniques developed lead to improvements in certain applications of the hyperbola method.

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

Bahman Saffari, R. C. Vaughan (1977)

Annales de l'institut Fourier

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.

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