Related papers: A Constructive Proof of Beal's Conjecture

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…

We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The…

In this paper, we study the non-existence of positive solutions for the following conformal $Q$-curvature equation \begin{equation*} (-\Delta)^\sigma u = K(x) u^{\frac{n+2\sigma}{n-2\sigma}} \quad \text{in } \mathbb{R}^n, \end{equation*}…

An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no…

We prove in most cases that a general smooth complete intersection in the projective space has no non-trivial automorphisms.

I show that the general implicit-function problem (or parametrized fixed-point problem) in one complex variable has an explicit series solution given by a trivial generalization of the Lagrange inversion formula. I give versions of this…

For a fixed integer a>1, we suggest that the probability of nullity of the p-Fermat quotient q(p,a) is much lower than 1/p for any arbitrary large prime number p. For this we use various heuristics, justified by means of numerical…

We consider the semilinear elliptic equation $$ -\Delta u = |x|^\alpha u^p \quad \hbox{in } \mathbb{R}^N, $$ where $N\ge 3$, $\alpha>-2$ and $p>1$. We show that there are no positive solutions provided that the exponent $p$ additionally…

The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.

In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and…

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. In this…

We show that solutions of the Bach equation exist which are not conformal Einstein spaces.

We give a generalized version of the Freyd conjecture and a way to think about a possible proof. The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new…

We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are $p$-divisible for infinitely many primes $p$, or…

The general solution to the Complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special…

In this paper, we show that any proof of the Collatz 3n+1 Conjecture must have an infinite number of lines; therefore, no formal proof is possible.

In this article we prove that linear differential equations with only algebraic solutions have zero $m$-curvature modulo $p^k$ for all except a finite number of primes $p$ and all $k,m\in\mathbb N$ with ${\rm ord}_pm!\geq k$. This provides…

Using elementary pcf, we show that there is no $j:V\to M,$ $M$ transitive, $j\lambda =\lambda >crit(j),$ $j^{\prime \prime}\lambda \in M.$

The existence of entire solutions to quadratic trinomial Fermat type differential-difference equations and \(q\)-difference differential equations involving second-order derivatives is studied by using Nevanlinna theory, and the exact form…

We prove that if all the simple groups involved in a finite group $G$ satisfy the `inductive Feit condition', then Walter Feit's conjecture from 1980 holds for $G$. In particular, this would solve Brauer's Problem 41 from 1963 in the…