Positive Integer Solutions Research Articles

The Diophantine equation (m2 + n2)x + (2mn)y = (m + n)2z

For fixed coprime positive integers [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text], there is a conjecture that the exponential Diophantine equation [Formula: see text] has only the positive integer solution [Formula: see text] for any positive integer [Formula: see text]. This is the analogue of Jésmanowicz conjecture. In this paper, we consider the equation [Formula: see text], where [Formula: see text] are coprime positive integers, and prove that the equation has no positive integer solution if [Formula: see text] and [Formula: see text].

Uniqueness of Solutions to Simultaneous Pell Equations

Let m, n be integers with $$1<m<n$$ , and let $$d=\gcd (m,n)$$ . In this paper, using the diophantine approximation method, we prove that if $$d \ge 2(n/d)^4$$ , then the simultaneous Pell equations $$x^2-(m^2-1)y^2=1$$ and $$z^2-(n^2-1)y^2=1$$ have only the positive integer solution $$(x,y,z)=(m,1,n)$$ .

On the Growth of Some Functions Related to z(n)

The order of appearance z : Z &gt; 0 → Z &gt; 0 is an arithmetic function related to the Fibonacci sequence ( F n ) n . This function is defined as the smallest positive integer solution of the congruence F k ≡ 0 ( mod n ) . In this paper, we shall provide lower and upper bounds for the functions ∑ n ≤ x z ( n ) / n , ∑ p ≤ x z ( p ) and ∑ p r ≤ x z ( p r ) .

Some results on natural numbers represented by quadratic polynomials in two variables

We consider a set of equations of the form Pj (x, y) = (10x + mj )(10y + nj ), x ≥ 0, y ≥ 0, j = 1,2,3, such that {m 1 = 7, n 1 = 3}, {m 2 = n 2 = 9} and {m 3 = n 3 = 1}, respectively. It is shown that if is a solution of the j’th equation one has the inequality , where and pj is a natural number ending in 1, such that , and hold, respectively. Moreover, assuming the previous result we show that , with , and , respectively. Finally, we present upper and lower bounds for the relevant positive integer solution of the equation defined by pj = (10A + mj )(10B + nj ), for each case j = 1, 2, 3, respectively.

A note on the exceptional solutions of Jeśmanowicz’ conjecture concerning primitive Pythagorean triples

Let $$(m,\ n)$$ be fixed positive integers such that $$m>n,\ \gcd (m,\ n)=1$$ and $$ mn\equiv 0 \pmod 2$$ . Then the triple $$(m^2-n^2,\ 2mn,\ m^2+n^2)$$ is called a primitive Pythagorean triple. In 1956, Jeśmanowicz (Wiadom Math 1(2):196–202, 1955/1956 ) conjectured that the equation $$(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$$ has only the positive integer solution $$(x,\ y,\ z)=(2,\ 2,\ 2)$$ . This problem is not solved yet. A solution $$(x,\ y,\ z)$$ of this equation is called exceptional if $$(x,\ y,\ z)\ne (2,\ 2,\ 2)$$ . In this paper, using Baker’s method, we prove that if $$m>\max \{10^{127550},\ n^{5127},\ n^{(\log n)^2}\}$$ , then the above equation has no exception solutions $$(x,\ y,\ z)$$ with $$ x\equiv y\equiv 0 \pmod 2 $$ . By this conclusion, we can deduce that if m, n satisfy the above condition, and $$ m\equiv 3 \pmod 4 $$ or $$ n\equiv 3 \pmod 4 $$ , then Jeśmanowicz’ conjecture is true.

On the Diophantine equation $$x^2+3^a41^b=y^n $$

In this paper we find all positive integer solutions (x, y, n, a, b) of the equation in the title for non negative integers a and b under the condition that the integers x and y are relatively prime and $$ n \ge 3$$ .

Linear combinations of prime powers in X-coordinates of Pell equations

Let $$\{X_\ell \}_{\ell \ge 1}$$ be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation $$X^2-dY^2=\pm 1$$ corresponding to a nonsquare integer $$d>1$$ . We show that there are only a finite number of nonsquare integers $$d > 1$$ such that there are at least two different elements of the sequence $$\{X_\ell \}_{\ell \ge 1}$$ that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents. Moreover, we solve explicitly the case in which two of the X-coordinates above are a sum of a power of two and a power of three under the above condition on the exponents. This work is motivated by the recent paper Bertok et al. (Int J Number Theory 13(02):261–271, 2017).

A conjecture of Erdős, supersingular primes and short character sums

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation \[ n(n+d)\cdots(n+(k-1)d) = y^\ell \] has at most finitely many solutions in positive integers $n$, $d$, $y$ and $\ell$, with $\mathrm{gcd}(n,d)=1$ and $\ell \ge 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.

Positive Integer Solutions of Some Diophantine Equations Involving Lucas-Balancing Numbers

Positive Integer Solutions of Some Diophantine Equations Involving Lucas-Balancing Numbers

An Elementary Proof of a Result Ma and Chen

In 1956, Je_smanowicz conjectured that, for positive integers <i>m</i> and <i>n</i> with <i>m</i> > <i>n</i>; gcd(<i>m</i>; <i>n</i>) = 1 and <i>m</i> ≢ <i>n</i> (mod 2), the exponential Diophantine equation (<i>m</i>² -<i>n</i>²)^x + (2<i>m</i><i>n</i>)^y = (<i>m</i>² + <i>n</i>²)^z has only the positive integer solution (<i>x</i>; <i>y</i>; <i>z</i>) = (2; 2; 2). Recently, Ma and Chen proved the conjecture if 4 <i>☓</i><i>m</i><i>n</i> and y ≥ 2. In this paper, we provide a proposition that, for positive integers <i>m</i> and <i>n</i> with <i>m</i> > <i>n</i>; gcd(<i>m</i>; <i>n</i>) = 1 and <i>m</i>² + <i>n</i>² ≡ 5 (mod 8), the exponential Diophantine equation (<i>m</i>² -<i>n</i>²)^x + (2<i>m</i><i>n</i>)^y = (<i>m</i>² + <i>n</i>²)^z has only the positive integer solution <i>x</i> = <i>y</i> = <i>z</i> = 2 with 2j gcd(<i>x</i>; <i>y</i>). Then we present an elementary and simple proof of the result of Ma and Chen by using Jacobi’s symbols.

On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$

Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.

Limits on Legs of Pythagorean Triples and Fermat’s Last Theorem

Are there any more elementary results for Fermat’s Last Theorem? Fermat’s Last Theorem states that there are no positive integer solutions to an+bn=cn for integer n greater than 2. The theorem was ...

An upper bound for the number of solutions of ternary purely exponential Diophantine equations II

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq 10^{62}$, then the equation has at most two positive integer solutions $(x,y,z)$.

An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples

Let $m$, $n$ be positive integers such that $m>n$, $\gcd(m,n)=1$ and $m \not\equiv n \bmod 2$. In 1956, L. Je\'smanowicz \cite{Jes} conjectured that the equation $(m^2 - n^2)^x + (2mn)^y = (m^2+n^2)^z$ has only the positive integer solution $(x,y,z) = (2,2,2)$. This problem is not yet solved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent \cite{Lau} with some elementary methods, we prove that if $mn \equiv 2 \bmod 4$ and $m > 30.8 n$, then Je\'smanowicz' conjecture is true.

Complete solutions of the simultaneous Pell’s equations x2 − (a2 − 1)y2 = 1 and y2 − pz2 = 1

Let [Formula: see text] be a positive integer with [Formula: see text], and let [Formula: see text] be an odd prime. In this paper, by using certain properties of Pell’s equations and quartic diophantine equations with some elementary methods, we prove that the system of equations [Formula: see text] [Formula: see text] and [Formula: see text] has positive integer solutions [Formula: see text] if and only if [Formula: see text] and [Formula: see text] satisfy [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are positive integers. Further, if the above condition is satisfied, then [Formula: see text] has only the positive integer solution [Formula: see text]. By the above result, we can obtain the following corollaries immediately. (i) If [Formula: see text] or [Formula: see text], then [Formula: see text] has no positive integer solutions [Formula: see text]. (ii) For [Formula: see text], [Formula: see text] has only the positive integer solutions [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

On some ternary pure exponential diophantine equations with three consecutive positive integers bases

By using the lower bound of linear forms in two logarithms of Laurent (Acta Arith. 133(4) (2008) 325–348), we give here a new solution that the ternary pure exponential diophantine equation $$(n+1)^{x}+(n+2)^{y}=n^{z}$$ has no positive integer solutions except for $$(n,x,y,z)=(3,1,1,2)$$ . This proof is very different from Le (J. Yulin Teachers College 28(3) (2007) 1–2), in which he used the classification method of solutions of exponential decomposition form equation. Furthermore, we solved completely another similar ternary pure exponential diophantine equation $$n^{x}+(n+2)^{y}=(n+1)^{z}$$ by using m-adic estimation of linear forms due to Bugeaud (Compos. Math. 132(2) (2002) 137–158).

On a class of quartic Thue equations with three parameters

Let k,m and n be integers. In this paper, for a fixed integer μ≠0, we show that the family of Thue equationx4−kmnx3y+(km2−kn2+2)x2y2+kmnxy3+y4=μ,is reducible by Tzanakis's method into a system of pellian equationskV2−(km2+4)U2=−4μ,kZ2−(kn2−4)U2=4μ,with any triple of integers (k,m,n) such that k>0, |n|≥2, |m|≥2. We consider this system for any even integer k≠2□, μ=1 and we prove that for all integers |n|≥2 and |m|≥2 that are sufficiently large and have sufficiently large common divisor this system has only the trivial solutions (U,V,Z,)=(±1,±m,±n). We also show that if k≠2□ is even, then the system has in general at most 8 solutions in positive integers.

Expressions for rational approximations to square roots of integers using Pell's equation

There are recursive expressions (see [1]) for sequentially generating the integer solutions to Pell's equation: p2 − Dq2 = 1, where D is any positive non-square integer. With known positive integer solution p1 and q1 we can compute, using these recursive expressions, pn and qn for all n &gt; 1. See Table in [2] for a list of smallest integer, or fundamental, solutions p1 and q1 for D≤ 128. These (pn, qn) pairs also form rational approximations to that, as noted in [3, Chapter 3], match with convergents (Cn = pn / qn) of the Regular Continued Fractions (RCF, continued fractions with the numerator of all fractions equal to 1) for .

On the equation x2 + ry2 = z2

In this Article, we derive a method for classifying all positive integer solutions of the equation x2 + ry2 = z2, where r is a given positive rational. In order to simplify notation, such a solution with x = a, y = b and z = c will be denoted by the ordered triple (a, b, c) and, in all that follows, the term solution will be taken to mean a positive integer solution of the above equation. The method employed will be similar to that used in [1] to find all integer triangles containing an angle whose cosine is known.

Newton Quadrilaterals, the Associated Cubic Equations, and Their Rational Solutions

A Newton quadrilateral is a convex quadrilateral inscribed in a circle, with one side a diameter. For diameter d and the other side lengths , Newton arrived at the cubic equation , which we shall refer to as Newton’s equation. Its positive integer solutions were found by P. Bachmann. All (positive and nonpositive) integer solutions were found by A. Oppenheim in terms of logarithmic and hyperbolic functions. A formula that produces infinitely many rational solutions was found by N. Anning, using an elegant relation among the cosines of the angles of a triangle. All rational solutions of Newton’s equation were found, using a short and elegant method based on linear algebra, by I. A. Barnett. The same results of Barnett were recently obtained independently by M. Hajja, using a relation among the cosines of the angles of what he called a generalized triangle. In the present paper, we use a direct algebraic method that also yields all rational solutions, albeit parameterized in a way very different from that obtained by Barnett (and Hajja), and we establish an explicit, nonobvious, birational correspondence between the two parameterizations. We also treat the nonconvex version of Newton’s problem and several related issues.