scholarly journals Preface to the special issue dedicated to combinatorics, automata and number theory

2008 ◽  
Vol 391 (1-2) ◽  
pp. 1-2
Author(s):  
Valérie Berthé ◽  
Pierre Lecomte ◽  
Michel Rigo
Keyword(s):  
Number Theory ◽  
Special Issue

scholarly journals Preface to the special issue on automatic sequences, number theory, and aperiodic order

2017 ◽  
Vol 28 (1) ◽  
pp. 1-2
Author(s):  
Valérie Berthé ◽  
Frits Beukers ◽  
Alex Clark ◽  
Robbert Fokkink ◽  
Cor Kraaikamp
Keyword(s):  
Number Theory ◽  
Special Issue ◽  
Automatic Sequences ◽  
Aperiodic Order

scholarly journals On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 288
Author(s):  
Atsushi Yamagami ◽  
Kazuki Taniguchi
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Number ◽  
Special Issue ◽  
Pascal’S Triangle ◽  
Mersenne Number ◽  
Modulo P ◽  
Pascal's Triangle ◽  
Positive Integers ◽  
Symmetric Property

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer k ≥ 2 . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the p e -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form p e m − 1 p e − 1 with some integer m ≥ 1 . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ( x + 1 ) p e ≡ ( x p + 1 ) p e − 1 ( mod p e ) of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.

Multiplicative Number Theory I

2006 ◽  
Author(s):  
Hugh L. Montgomery ◽  
Robert C. Vaughan
Keyword(s):  
Number Theory
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