相关论文: Orthogonal matrices with rational components in co…
Even though flt is a number theoretic result we prove that the result depends on the topological as well as the field structure of the underlying space.
Modern mathematics is known for its rigorous proofs and tight analysis. Math is the paradigm of objectivity for most. We identify the source of that objectivity as our knowledge of the physical world given through our senses. We show in…
We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also…
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational…
In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the…
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…
This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were…
The received Hilbert-style axiomatic foundations of mathematics has been designed by Hilbert and his followers as a tool for meta-theoretical research. Foundations of mathematics of this type fail to satisfactory perform more basic and more…
Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates…
One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those…
The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$…
In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices,…
An age-old controversy in mathematics concerns the necessity and the possibility of constructive proofs. The controversy has been rekindled by recent advances which demonstrate the feasibility of a fully constructive mathematics. This…
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real,…
Let $K$ be a totally real field. By the asymptotic Fermat's Last Theorem over $K$ we mean the statement that there is a constant $B_K$ such that for prime exponents $p>B_K$ the only solutions to the Fermat equation $a^p + b^p + c^p = 0$…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of…