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Fermat’s Little Theorem via Divisibility of Newton’s Binomial

Rafał Ziobro (2015)

Formalized Mathematics

Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in...

Fibonacci Numbers with the Lehmer Property

Florian Luca (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that if m > 1 is a Fibonacci number such that ϕ(m) | m-1, where ϕ is the Euler function, then m is prime

Finite and periodic orbits of shift radix systems

Peter Kirschenhofer, Attila Pethő, Paul Surer, Jörg Thuswaldner (2010)

Journal de Théorie des Nombres de Bordeaux

For r = ( r 0 , ... , r d - 1 ) ∈ ℝ d define the function τ r : ℤ d → ℤ d , z = ( z 0 , ... , z d - 1 ) ↦ ( z 1 , ... , z d - 1 , - rz ) , where rz is the scalar product of the vectors r and z . If each orbit of τ r ends up at 0 , we call τ r a shift radix system. It is a well-known fact that each orbit of τ r ends up periodically if the polynomial t d + r d - 1 t d - 1 + ⋯ + r 0 associated to r is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of...

F-Normalreihen.

Herbert Möller (1977)

Journal für die reine und angewandte Mathematik

Fonctions digitales le long des nombres premiers

Bruno Martin, Christian Mauduit, Joël Rivat (2015)

Acta Arithmetica

In a recent work we gave some estimations for exponential sums of the form ∑ n ≤ x Λ ( n ) e x p ( 2 i π ( f ( n ) + β n ) ) , where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.

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