相关论文: A report on Wiles' Cambridge lectures
One shows that the Last Fermat Theorem is equivalent to the statement that all rational solutions of the famous equation are provided by an orbit of rationally parametrized subgroup of a group preserving k-ubic form. This very group…
An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…
We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather…
These lecture notes have been converted to a book titled Network Information Theory published recently by Cambridge University Press. This book provides a significantly expanded exposition of the material in the lecture notes as well as…
In our paper from 1985 we have constructed two integrals of the Riemann's function $Z^2(t)$ over two disconnected sets with asymptotically equal measures such that these two integrals differ by considerably big excess. In the present paper…
Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.
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…
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…
Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from…
In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of…
We give a new proof of a_4\phi_3 summation due to G.E. Andrews and confirm another_4\phi_3 summation conjectured by him recently. Some variations of these two_4\phi_3 summations are also given.
`Fermat's Last Theorem for the exponent 3 has received numerous proofs, the most common of which being either in Euler's or in Gauss' style. This latter works entirely in the ring of integers of the quadratic field generated by the square…
The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves…
In this paper we give some new consequences that follow from our formula for increments of the Hardy-Littlewood integral. The main of these ones are $\mathcal{T}_1$ and $\mathcal{T}_2$ equivalents of the Fermat-Wiles theorem.
From some works of P. Furtw\"angler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for…
Fermat's statement is equivalent to say that if $x$, $y$, $z$, $n$ are integers and $n>2$, then $z^{n}\gtrless x^{n}+y^{n}$. This is proved with the aid of numbers $\lambda $'s, of the form $\lambda =z/\rho $, with $1<\rho<z$, named…
This text is based on a series of three expository lectures on a variety of topics related to "thin orbits," as delivered at Durham University's Easter School on "Dynamics and Analytic Number Theory" in April 2014. The first lecture reviews…
We prove, in the context of the section conjecture, that every Selmer section over $\mathbb{Q}$ of the affine Fermat curve with exponent $\ell$ is cuspidal for $\ell\geq 7$.
Let $F$ be a finite field, $\mu$ be a fixed additive character and $s$ be an integer coprime to $|F^\times|$. For any $a\in F$, the corresponding Weil sum is defined to be $W_{F,s}(a)=\displaystyle\sum_{x \in F} \mu(x^s-ax)$. The Weil…
The past few years have produced major advances in our understanding of the quantum Hall effects---quantized and unquantized. Theories based on a mathematical transformation, where the electrons are replaced by a set of fermions interacting…