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Short remark on Fibonacci-Wieferich primes

Jiří Klaška (2007)

Acta Mathematica Universitatis Ostraviensis

This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime p is Fibonacci-Wieferich is equal to 1 / p . According to our computational results and some theoretical consideratons, another form of probability can...

Some Borel measures associated with the generalized Collatz mapping

K. Matthews (1992)

Colloquium Mathematicae

This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers ℤ ^ and construct finitely many ergodic Borel measures on ℤ ^ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.

Subsequence sums of zero-sum free sequences over finite abelian groups

Yongke Qu, Xingwu Xia, Lin Xue, Qinghai Zhong (2015)

Colloquium Mathematicae

Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that | Σ ( X ) | ≥ 2 r - 1 ( | X | - r + 2 ) - 1 for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

We describe all sets A ⊆ p which represent the quadratic residues R ⊆ p in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

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