Related papers: Windmills of the minds: an algorithm for Fermat's …
Let $\alpha(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $\alpha(2n+1) = (2n+1) \alpha(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents…
This note concerns exponential sheaves and the "universal" Fourier transform on them. Fourier invertibility and the subsequent Fourier miracle is demonstrated. Further, t-structures and realizations are constructed and shown to have…
Using the matrix-forest theorem and the Parisi-Sourlas trick we formulate and solve a one-matrix model with non-polynomial potential which provides perturbation theory for massive spinless fermions on dynamical planar graphs. This is a…
The Titchmarsh convolution theorem is a celebrated result about the support of the convolution of two functions. We present a simple proof based on the canonical factorization theorem for bounded holomorphic functions on the unit disk.
The association of information with entropy has been argued on plausibility arguments involving the operation of imaginary engines and beings, and it is not a universal theorem. In this paper, a theorem by Charles Bennett on reversible…
A Turmit is a Turing machine that works over a two-dimensional grid, that is, an agent that moves, reads and writes symbols over the cells of the grid. Its state is an arrow and, depending on the symbol that it reads, it turns to the left…
Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the…
This research focuses on the Numerical approach for Fermat's Last theorem. We can induce an Alternative form of Fermat's last theorem by using particular geometric mapping $\mathcal{M}$ on a Cartesian plane to a Torus. It transforms the…
In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.
The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological…
The problem of simplicity of Fermat number-twins $f_{n}^{\pm}=2^{2^n}\pm3$ is studied. The question for what $n$ numbers $f_{n}^{\pm}$ are composite is investigated. The factor-identities for numbers of a kind $x^2 \pm k $ are found.
A new density matrix and corresponding quantum kinetic equations are introduced for fermions undergoing coherent evolution either in time (coherent particle production) or in space (quantum reflection). A central element in our derivation…
We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Tribonacci-automatic". This class includes, for example, the famous Tribonacci word T =…
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 present the only proof of Pierre Fermat by descente infinie that is known to exist today. As the text of its Latin original requires active mathematical interpretation, it is more a proof sketch than a proper mathematical proof. We…
In this paper we compute certain two-point integrals over a moduli space of stable maps into projective space. Computation of one-point analogues of these integrals constitutes a proof of mirror symmetry for genus-zero one-point…
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of…
In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary…
Permutations can be viewed as pairs of linear orders, or more formally as models over a signature consisting of two binary relation symbols. This approach was adopted by Albert, Bouvel and F\'eray, who studied the expressibility of…
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…