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In this note a far extension of the Banach fixed point theorem is proved.

We give a proof of the Marker-Steinhorn Theorem which fills a gap in previous proofs of the result.

We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.

This article presents simple and easy proofs of the Implicit Function Theorem and the Inverse Function Theorem, in this order, both of them on a finite-dimensional Euclidean space, that employ only the Intermediate Value Theorem and the…

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

This is a survey of old and new problems and results in additive number theory.

The paper contains an alternative proof of M. Kontsevich Formality Theorem.

The recently developed proof of Fermat's Last Theorem is very lengthy and difficult, so much so as to be beyond all but a small body of specialists. While certainly of value in the developments that resulted, that proof could not be, nor…

We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.

In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.

This note presents a unified theorem of the alternative that explicitly allows for any combination of equality, componentwise inequality, weak dominance, strict dominance, and nonnegativity relations. The theorem nests 60 special cases,…

This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without…

In our work we give the examples using Fermat's Last Theorem for solving some problems from algebra and number theory.

This paper gives a self-contained group-theoretic proof of a dual version of a theorem of Ore on distributive intervals of finite groups. We deduce a bridge between combinatorics and representations in finite group theory.

We show that an elementary proof of Fermat's Last Theorem (FLT) exists. Our paper also extends the scope of FLT from integers to all rational numbers.

In this paper, we prove a fine condensation theorem. This is quite similar to condensation theorems for pure extender mice in the literature, except that condensation for iteration strategies has been added to the mix.

Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.

This paper presents a novel direct elementary proof for Fermat's Last Theorem. We use algebra, modular math, and binomial series to develop inherent mathematical relationships hidden within Fermat's Last Theorem. With these derived…

We prove a uniqueness theorem for an entire function, which shares certain values with its higher order derivatives.

We present a simple proof of the fundamental theorem of Galois theory, which establishes a correspondence between the intermediate fields of a finite Galois extension and the subgroups of its Galois group. The proof is based on the…