Diophantine Equations Research Articles

Diophantine equations on cross-multiplicative forms of the Pell and Pell-Lucas numbers

This study aims to explore all Pell numbers that are the product of two random Pell-Lucas numbers and all Pell-Lucas numbers that are the product of two random Pell numbers based on linear forms in logarithms of algebraic numbers using Matveev's theorem and Dujella - Pethő reduction lemma. Further, we find all the common terms of Pell and Pell-Lucas numbers and show that no Pell and no Pell-Lucas numbers can be written as a square of another.

Equations in wreath products

Abstract We survey solvability of equations in wreath products of groups, and prove that the quadratic diophantine problem is solvable in wreath products of Abelian groups. We consider the related question of determining commutator width, and prove that the quadratic diophantine problem is also solvable in Baumslag’s finitely presented metabelian group. This text is a short version of an extensive article by the first-named authors.

On the largest value of the solutions of Erdős’ last equation

Let [Formula: see text] be a positive integer. The Diophantine equation [Formula: see text] is called Erdős’ last equation. We prove that [Formula: see text] as [Formula: see text] and determine all tuples [Formula: see text] with [Formula: see text].

On the generalization neo balancing sequence and some applications

We investigate the generalization of sequence neo balancing numbers and their recurrence relations by extending the catalyst of certain sequences in balancing numbers to any integer k in the sequence neo balancing numbers. We derive the Diophantine equation for sequence neo balancing numbers in terms of k, which corresponds to the Diophantine equation for neo balancing numbers. We derive the Diophantine equation for the sequence neo balancing numbers and solve it via Pell's equation and Brahmagupta's identity. We examine the square root term in the derived Diophantine equation for the sequence neo balancing numbers by treating it as the generalized Pell's equation. Simultaneously, we consider the well-known Pell's equation. We integrate the generalized Pell's equation for the sequence neo balancing numbers to the well-known Pell's equation by using the Brahmagupta's identity. We obtain two solutions for both the generalized Pell's equation for the sequence neo balancing numbers and the well-known Pell's equation. The obtained solutions are sometimes analogous for some values of k. Then we investigate more precisely each case and substitute the solutions into the derived Diophantine equation for the sequence neo balancing numbers. Therefore, there are values of k that make both solutions analogous implying the recurrence relation can give all terms in the sequence neo balancing numbers. Simultaneously, there are values of k that make both solutions different implying that we need two recurrence relations generated by the two solutions to complete the sequence neo balancing numbers. Moreover, we establish a few theorems to explain why some values of k generate similar sequence of neo balancing numbers.

Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b

Let (Pk)k≥0 be a Padovan sequence and (Rk)k≥0 be a Perrin sequence. Let n, m, b, and k be non-negative integers, where 2≤b≤10. In this paper, we are devoted to delving into the equations Rn=bdPm+Rk and Rn=bdRm+Pk, where d is the number of digits of Rk or Pk in base b. We show that the sets of solutions are Rn∈{R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R19,R23,R25,R27} for the first equation and Rn∈{R0,R2,R3,R4,R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R18,R20,R21} for the second equation. Our approach employs advanced techniques in Diophantine analysis, including linear forms in logarithms, continued fractions, and the properties of Padovan and Perrin sequences in base b. We investigate both the deep structural symmetries and the complex structures that connect recurrence relations and logarithmic forms within Diophantine equations involving special number sequences.

POLYNOMIALS REPRESENTED BY NORM FORMS VIA THE BETA SIEVE

Abstract A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation $f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$ . Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.

The Diophantine equation $$b (b+1) (b+2) = t a (a + 1) (a + 2)$$ and the gap principle

The Diophantine equation $$b (b+1) (b+2) = t a (a + 1) (a + 2)$$ and the gap principle

Rational spherical triangles

A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of [Formula: see text]. Little and Coxeter have given examples of rational spherical triangles in the 1980s. In this work, we are interested in determining all the rational spherical triangles. We introduce a conjecture on the solutions to a trigonometric Diophantine equation. An implication of the conjecture is that the only rational spherical triangles are the ones given by Little and Coxeter. We prove some partial results towards the conjecture.

On solutions of some Diophantine equations

Let k and p be prime numbers with p>2. In this note, we find all positive integer solutions to the Diophantine equations x^2-kxy-y^2=±k, x^2-(k^2+4) y^2=±4k, x^2-2pxy-y^2=±2p, x^2-kxy-y^2=±k(k^2+4) in terms of generalized Fibonacci and Lucas numbers.

Class fields and form class groups for solving certain quadratic Diophantine equations

Class fields and form class groups for solving certain quadratic Diophantine equations

On the resolution of the Diophantine equation $$U_n + U_m = x^q$$

On the resolution of the Diophantine equation $$U_n + U_m = x^q$$

Fermat's Last Theorem and the Golden Mean

We attempt to find solutions to the Diophantine equations from Fermat's Last Theorem in the ring Z[tau], where tau is the golden mean. We begin with the case when n=3 and create an algorithm to generate solutions to the equation. Out of these solutions, we have found only four to be primitive. Also, we attempt to find solutions under higher powers greater than three but have not found any solutions in such cases. The algorithm and solutions themselves are all provided in the paper.

On the Diophantine equation $~(p^n)^x+(4^mp+1)^y=z^2~$ when $~p, 4^mp+1~$ are prime numbers

In this paper, we solve the Diophantine equation $(p^n)^x+(4^mp+1)^y=z^2$ in $\mathbb{N}$ for $p\geq 3$ and $1+4^mp$ are prime integers. Concretely, using the congruent method, we prove that this equation has no non-negative solutions if $p>3$. For the case $p=3$, we will show that this equation has no solutions if $m>1$. Furthermore, in this case, when $m=1$ using the elliptic curves, we will show that this equation has only solution $(x,y,z)=(3, 2, 14)$ if $n=1$ and $(x,y,z)=(1,2,14)$ if $n=3$.

Minimisation of 2-coverings of genus 2 Jacobians

An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and reduction. We give an algorithm for minimising certain pairs of quadratic forms, subject to the constraint that the first quadratic form is fixed. This has applications to 2-descent on the Jacobian of a genus 2 curve.

On Extension of Existing Results on the Diophantine Equation: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\)

Let $w_r$ be a given sequence in arithmetic progression with common difference $d$. The study of diophantine equation, which are polynomial equations seeking integer solutions has been a very interesting journey in the field of number theory. Historically, these equations have attracted the attention of many mathematicians due to their intrinsic challenges and their significance in understanding the properties of integers. In this current study we examine a diophantine equation relating the sum of square integers from specific sequences to a variable $d$. In particular, on extension of existing results on the diophantine equation: $\sum_{r=1}^{n} w^2_r +\frac{n}{3}d^2= 3(\frac{nd^2}{3} +\sum^{\frac{n}{3}}_{n=1} w^{2}_{3r-1})$ is introduced and partially characterized.

Conjunctions of exponential diophantine equations over $${\mathbb {Q}}$$

Conjunctions of exponential diophantine equations over $${\mathbb {Q}}$$

A Discussion on the Solution(s) of the Diophantine Equation 3<sup>x</sup> + 15<sup>y</sup> =Z<sup>2</sup>

The absence of a generalized method for solving Diophantine equations having more unknowns than a number of equations is a challenge for researchers in different fields. The presence of the Diophantine equation is reported in the study of the Hydrogen spectrum, quantum Hall effect, chemistry, cryptography, etc. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation 3x + 15y =z2 is addressed in this paper to find the solution(s) in positive integers. It is found that the equation has only two solutions of (x,y,z) as (1,0,2) and (0,1,4) in non-negative integers.

On equal values of products and power sums of consecutive elements in an arithmetic progression

In this paper, we study the Diophantine equation $$b^k+(a+b)^k+(2a+b)^k+\cdots+(a(x-1)+b)^k=y(y+c)(y+2c)\cdots(y+(\ell-1)c),$$ where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$, and obtain a general ineffective result based on the Bilu--Tichy method.

Integer Solutions to Quadratic Diophantine Equations using Efficient Algorithms with Elementary and Quadratic Ring Methods

In this paper, we study the solvability of quadratic Diophantine equations x2 - Dy2 =N, where x and y are unknown integers, and D is a positive integer that is a square free and N is a nonzero integer. We use elementary and quadratic ring methods to find integer solutions of these equations. These methods involve concepts like units, fundamental units, norms, and conjugates in quadratic rings. We propose efficient algorithms to solve the equations for cases where |N| > \sqrt{D} and |N| < \sqrt{D}. The algorithms include the continued fraction algorithm, periodic quadratic algorithm, Lagrange-Matthew-Mollin algorithm, and brute-force search. These algorithms can be implemented in programming languages. Finally, we compare the algorithms and analyze their time complexity.

On the Diophantine Equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ Regarding Terai's Conjecture

This study establishes that the sole positive integer solution to the exponential Diophantine equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ is $(x,y,z)=(1,1,2)$ for all $r>1$. The proof employs elementary techniques from number theory, a classification method, and Zsigmondy's Primitive Divisor Theorem.