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Factors of a perfect square

Tsz Ho Chan (2014)

Acta Arithmetica

We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between √ n - ∜ n ( l o g n ) 1 / 7 and √ n + ∜ n ( l o g n ) 1 / 7 .

Fermat k -Fibonacci and k -Lucas numbers

Jhon J. Bravo, Jose L. Herrera (2020)

Mathematica Bohemica

Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all k -Fibonacci and k -Lucas numbers which are Fermat numbers. Some more general results are given.

Finiteness of odd perfect powers with four nonzero binary digits

Pietro Corvaja, Umberto Zannier (2013)

Annales de l’institut Fourier

We prove that there are only finitely many odd perfect powers in ℕ having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at S -unit points (in a suitable ν -adic convergence), Roth’s...

Fonction ζ de Carlitz et automates

Valérie Berthé (1993)

Journal de théorie des nombres de Bordeaux

Carlitz a défini sur 𝔽 q une fonction ζ et une série formelle I I , analogues respectivement à la fonction ζ de Riemann et au réel π . Yu a montré, en utilisant les modules de Drinfeld, que ζ ( s ) / I I 3 est transcendant pour tout s non divisible par q - 1 . Nous donnons ici une preuve «automatique» de la transcendance de ζ ( s ) / I I 3 pour 1 ≤ s ≤ q - 2 , en utilisant le théorème de Christol, Kamae, Mendès France et Rauzy.

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