Related papers: A note on lower bounds for numerical series
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction,…
In this note, we shall give an improved lower bound for the argument of a power of a given algebraic number which has absolute value one but is not a root of unity.
Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…
We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications…
We present an elementary proof of the fundamental theorem of algebra, following Cauchy's version but avoiding his use of circular functions. It is written in the same spirit as Littlewood's proof of 1941, but reduces it to more elementary…
We prove logarithmic lower bounds on digital sums of powers, multiples of powers, factorials, and the least common multiple of $\{1,\ldots, n\}$, using only elementary number theory. We conclude with an expository proof of Stewart's theorem…
We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is…
In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in…
This set of notes re-proves known results on weighted automata (over a field, also known as multiplicity automata). The text offers a unified view on theorems and proofs that have appeared in the literature over decades and were written in…
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.
We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…
This note imparts heuristic arguments and theorectical evidences that contradict the abc conjecture over the rational numbers. In addition, the rudimentary datails for transforming this problem into the doimain of equidistribution theory…
In this short note, we give an affirmative answer to Wu's conjecture on practical numbers, which was posed in [X.-H. Wu, {\it Special forms and the distribution of practical numbers}, Acta Math. Hungar., {\bf 160}(2020), 405-411].
In this expository note we present simple proofs of the lower bound of Ramsey numbers (Erd\"os theorem), and of the estimation of discrepancy. Neither statements nor proofs require any knowledge beyond high-school curriculum (except a minor…
This short note contains elementary evaluations of some Euler sums.
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…