Related papers: A new proof of Lucas' Theorem

We state and prove a Lemma in 1 variable Calculus, that justifies some arguments previously used to ilustrate non-uniqueness of some generalized physical quantities.

We revisit Ahlfors theory of covering surfaces thanks to Stokes theorem.

Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = \alpha_i\alpha_{i-1}\dots \alpha_0$ and $B = \beta_i\beta_{i-1}\dots \beta_0$. Lucas proved that $$\binom{A}{B} \equiv…

We give a new proof of a theorem of B.M. Bredihin which was originally proved by extending Linnik's solution, via his dispersion method, of a problem of Hardy and Littlewood.

We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.

In this paper we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev…

In this paper, we show the new fixed point theorem in metric spaces. Furthermore, for this fixed point theorem, we apply to the Collatz conjecture.

We use actions by finite cyclic groups to derive generalizations of three classical theorems from elementary number theory.

We modify the proof of the basic lemma of a paper of Saks and Zygmund on additive functions of rectangles.

We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.

In this note, we present a simple non-directed graph proof of Sharkovsky's theorem which is different from the one given in [2].

A new generalization of the classical separate algebraicity theorem is suggested and proved.

In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions.…

Lov\'asz gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a different short proof by reducing to the cubic case. Then we show how to extend the result to (online) list coloring via the Kernel Lemma.

We present an elementary proof of Fermat's Last Theorem. No ancillary results are used, not even the most basic ones. The proof directly leads to a contradiction of the Fermat equation in the set of integers.

In their study of a binomial sum related to Wolstenholme's theorem, Chamberland and Dilcher prove that the corresponding sequence modulo primes $p$ satisfies congruences that are analogous to Lucas' theorem for the binomial coefficients…

This article gives an alternative proof of the fact that N_{Q(zeta)/Q}(1-zeta)=p where p is an odd prime number and zeta is a primitive p-th root of unity, and uses it to prove that N_{Q(zeta)/Q}(1+zeta-zeta^2)=L(p) the p-th Lucas number.…

We present a new proof of the classical divergence theorem in bounded domains. Our proof is based on a nonlocal analog of the divergence theorem and a rescaling argument. Main ingredients in the proof are nonlocal versions of the divergence…

In this paper we give a new proof of Riemann's well known mapping theorem. The suggested method permits to prove an analog of that theorem for the three dimensional case.