arXiv subjects
Publications and source records attributed to Felix Sidokhine.
We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a positive solution.
Shnirelman's theorem is applied to solving Diophantine equations, and also discussing of the problems of a representation of Gaussian integers by a sum of odd Gaussian primes.
In this paper we propose an alternative formulation of the binary and ternary Goldbach conjectures as the systems of equations involving the Euler $ϕ$-function.
In our work we study the equations of the form $aX^4+bX^2 Y^2+cY^4=dZ^2$ over Gaussian integers by a method of the resolvents. We study as a new equations $X^4+6X^2 Y^2+Y^4=Z^2$ (Mordell's equation over $\mathbb{Z}[i]$), $X^4+6(1+i)X^2Y^2+2iY^4=Z^2$ and $X^4\pm Y^4=(1+ i)Z^2$ and give the new proofs of the known theorems on $X^4+Y^4=Z^2$ (Fermat - Hilbert), $X^4\pm Y^4=iZ^2$ (Szabó - Najman).
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
In our work we give the examples using Fermat's Last Theorem for solving some problems from algebra, geometry and number theory
In our work we give the examples using Fermat's Last Theorem for solving some problems from algebra and number theory.
The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.
Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+ε}p_n)$ where $ε> 0$, in the context of the results we have obtained in our paper.
In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems, including Ingham's exponential theorem as well as some new results. Diophantine inequalities and their connection with Cramer's and Andrica's conjectures are also discussed.
In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ represents all gaussian integers. The main tools used in this proof are Fermat's little theorem (over finite field extensions), the Mordell-Niven theorem (representation of some gaussians), and the generalized Euler-identity over finite field extensions.
In our notice we propose the classification of some quartic equations with only trivial solutions by the auxiliary equations. For proving trivial solutions of the quartic equations we use method infinite descent based on the number of prime integers of the solution of the auxiliary equation. The quartic equations with only trivial solutions belong to "higher arithmetic". The main tools of their investigation are the fundamental theorem of arithmetic, infinite descent and elliptic curves. In 1969, L.J. Mordell has observed that the triviality of the solution of $x^4 + y^4 = 2z^2$ implies the triviality of the solution of $x^4 + 6x^2y^2 + y^4 = z^2$. In his treatise, he did not study the reason or the mechanics of this relationship nor did he conjecture that this link extended to other fourth-degree diophantine equations. We have discovered that the connection identified by Mordell can be described by using the concept "resolvents". In present notice which carries an expository character, we show our findings by presenting the reader with a very easy to follow and transparent example of how the proposed technique works.
The goal of this notice is to present a proof of Bachet's conjecture based exclusively on the fundamental theorem of arithmetic. The novelty of this proof consists in its introduction of a partial order on rational integers through the unique factorization property. In general, the proofs of Bachet's conjecture by Lagrange - Euler's method assume necessary the use of infinite descent. In the proposed proof we do not assume the existence of a "minimal solution", but rather we show the existence of the desired solution through an algorithmic method.