A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples.
A set of $m$ distinct nonzero rationals $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+1$ is a perfect square for all $1\leq i<j\leq m$, is called a rational Diophantine $m$-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in …
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and Dujella, Kazalicki, Mikic and Szikszai recently proved that there exist infinitely many rational Diophantine sextuples. In this paper, generalizing the work of Piezas, we describe a method for generating new parametric formulas for …
A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm used to find them is described. A classification of quadruples and quintuples with examples and statistics is also given.
A set of $m$ positive integers $\{a_1, a_2, \dots , a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i < j \le m$. In 2004 Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$-tuples states that no Diophantine quintuple exists at all. In this paper we prove this conjecture.
We show that for infinitely many square-free integers q there exist infinitely many triples of rational numbers {a, b, c} such that a^2 + q, b^2 + q, c^2 + q, ab + q, ac + q and bc + q are squares of rational numbers.
We prove that for every integer $n$, there exist infinitely many $D(n)$-triples which are also $D(t)$-triples for $t\in\mathbb{Z}$ with $n\ne t$. We also prove that there are infinitely many triples with the property $D(-1)$ in $\mathbb{Z}[i]$ which are also $D(n)$-triple in $\mathbb{Z}[i]$ for two distinct $n$'s other than $n = -1$ and these triples are not equivalent to any triple with the property $D(1)$.
In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are fixed arbitrary nonzero integers. By our method, we may find infinitely many nontrivial positive solutions and also obtain infinitely many nontrivial parametric solutions for the Diophantine equations for every arbitrary integers $n$, $m$, $a_i$ and $b_i$.
In this paper we obtain several parametric solutions of the quartic diophantine equation $(x_1^4+x_2^4)(y_1^4+y_2^4)=z_1^4+z_2^4$. We also show how infinitely many parametric solutions of this equation may be obtained by using elliptic curves.
A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, \ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to those of the seminal work of B. J. Birch, for which the equation $F (x_1, \ldots, x_n) = 0$ has infinitely many integer solutions with semiprime coordinates. Previously it was known, by a result of Á. Magyar and T. Titichetrakun, that under the same hypotheses there exist infinitely many integer solutions to the equation with coordinates that have at most $384 n^{3…
In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an app…
Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $σ_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\leq i\leq n$ the system of Diophantine equations \begin{equation*} σ_{i}(\bar{X}_{2n})=a, \quad σ_{2n-i}(\bar{X}_{2n})=b, \quad σ_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. …
We show that for any irrational $α$ and any $τ<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|nα\|\leq n^{-τ}.$$ We also show that for all sufficiently large $b$ there exist 3-digit palindromes in base $b$ with precisely two prime factors.
Let Q be an infinite set of positive integers. Denote by W(Q) the set of n-tuples of real numbers simultaneously tau-well approximable by infinitely many rationals with denominators in Q but only by finitely many rationals with denominators in the complement of Q. The Hausdorff dimension of the liminf set W(Q) is computed when tau > 2 + 1/n. A p-adic analogue of the problem is also studied.
We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3. These results rely on exhibiting families of positive-rank elliptic curves.
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $ S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $ S$ is an $(F,2)$-Diophantine set. In addition, we show…
If $V$ is an irreducible algebraic variety over a number field $K$, and $L$ is a field containing $K$, we say that $V$ is diophantine-stable for $L/K$ if $V(L) = V(K)$. We prove that if $V$ is either a simple abelian variety, or a curve of genus at least one, then under mild hypotheses there is a set $S$ of rational primes with positive density such that for every $\ell \in S$ and every $n \ge 1$, there are infinitely many cyclic extensions $L/K$ of degree $\ell^n$ for which $V$ is diophantine-stable. We use this result to study the collection of finite extensions of $K$ generated by points in…
Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This provides an analogue to the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that a…
We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$
A Piatetski-Shapiro sequence with exponent $α$ is a sequence of integer parts of $n^α$ $(n = 1,2,\ldots)$ with a non-integral $α> 0$. We let $\mathrm{PS}(α)$ denote the set of those terms. In this article, we study the set of $α$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(α)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $α> 2$ such that $\mathrm{PS}(α)$ contains infinitely many arithmetic progressions of length $3$.
In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\,…
We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions $n,b,a_i\in\mathbb{Z}^+,i=1,2,\ldots,s-1,s\geq 3.$ By using the theory of elliptic curves, we prove that it has no solutions $n,b,a_i\in\mathbb{Z}^+$ for $s=3$, but for $s=4$ it has infinitely many solutions $n,b,a_i\in\mathbb{Z}^+$ and for $s\geq 5$ there are infinitely many polynomial solutions $n,b,a_i\in\mathbb{Z}[t_1,t_2,\ldots,t_{s-3}]$ with positive value satisfying this Diop…
In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for $(p,q,r,s)\in\{(2,4,8,8)…
In this paper, we solve the simultaneous Diophantine equations(SDE) x_1^u+...+x_n^u=k(y_1^u+...+y_{n/k}); u=1,3, where n >3, and k< n, is a divisor of n , and obtain nontrivial parametric solution for them. Furthermore we present a method for producing another solution for the above Diophantine equation (DE) for the case u = 3, when a solution is given. We work out some examples and find nontrivial parametric solutions for each case in nonzero integers. Also we prove that the other DE p_1x_1^{a_1}+....+p_nx_n^{a_n}=q_1y_1^{b_1}+...+q_my_m^{b_m} , has parametric solution and infinitely many sol…
Quartic Rational Diophantine Quadruples and the Euler Surface
University of Zagreb, Faculty of Civil Engineering（萨格勒布大学土木工程学院）；University of Zagreb, Faculty of Science（萨格勒布大学理学院）；University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）
In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \[\begin{cases} \begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\equiv0\pmod{(m^2+n^2)^2},\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\equiv0 \pmod{(m^2+n^2)^2}, \end{split} \end{cases}\] where $k\equiv0\pmod{4}$ when…
We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.
Let $β>1$ be a real number and define the $β$-transformation on $[0,1]$ by $T_β:x\mapsto βx\bmod 1$. Further, define $$W_y(T_β,Ψ):=\{x\in [0, 1]:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\}$$ and $$W(T_β,Ψ):=\{(x, y)\in [0, 1]^2:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\},$$ where $Ψ:\mathbb{N}\to\mathbb{R}_{>0}$ is a positive function such that $Ψ(n)\to 0$ as $n\to \infty$. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain serie…
Let $α$ and $β$ be real numbers such that $1$, $α$ and $β$ are linearly independent over $\mathbb{Q}$. A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+αx_1+βx_2| < \max\{|x_1|,|x_2|\}^{-2}$. In 1976, W. M. Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $γ\cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb{Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + αx_1 + βx_2| \max\{|x_1|,|x_2|\}^γ$ is arbitrarily small. Although Sc…
A cover of normal varieties is exceptional over a finite field if the map on points over infinitely many extensions of the field is one-one. A cover over a number field is exceptional if it is exceptional over infinitely many residue class fields. The first result: The category of exceptional covers of a normal variety, Z, over a finite field, F_q, has fiber products, and therefore a natural Galois group (with permutation representation) limit. This has many applications to considering Poincare series attached to diophantine questions. The paper follows three lines: * The historical role of th…
In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let $T_β$ be the beta-transformation with base $β>1$, $\{x_{n}\}_{n\geq 1}$ be a sequence of real numbers in $[0,1]$ and $φ\colon \mathbb{N}\rightarrow (0,1]$ be a positive function. With a detailed analysis on the distribution of {\em full cylinders} in the base space $\{β>1\}$, it is shown that for any given $x\in(0,1]$, for almost all or almost no bases $β>1$, the orbit of $x$ under $T_β$ can $φ$-well approximate the sequence $\{x_{n}\}_{n\geq 1}$ a…
Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a system S \subseteq E_n such that S has infinitely many integer solutions and each integer tuple (x_1,...,x_n) that solves S satisfies x_1=f(n). For each integer n>=12 we construct a system S \subseteq E_n such that S has infinitely many integer solutions and they all belong to Z^n\[-2^{2^{n-1}},2^{2^{n-1}}]^n.
Let $σ_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots <k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai \cite{ZC,ZC2}. More precisely, we prove that for each $n\geq 4$ and rational numbers $a, b$ with $ab\neq 0$, the system of diophantine equations \begin{equation*} σ_{1}(x_{1},\ldots, x_{n})=a, \quad σ_{n}(x_{1},\ldots, x_{n})=b, \end{equation*} has infinitely many solutions depending on $n-3$ free parameters. A similar result is proved for the system \beg…
Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $ψ$ let $ W_{\cal A}(ψ)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely many $(p,q) \in \Z \times {\cal A} $ such that $ |x - p/q| < ψ(q) $. The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for $ W_{\cal A}(ψ) \cap K $. One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in $K$ -- an assertion attributed to K. Mahler.
Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(ψ_n)_{n \in \mathbb{N}}$ and differentiable functions $(φ_n: J \to \mathbb{R})_{n \in \mathbb{N}}$ so that for Lebesgue-a.e. $θ\in J$, the inequality $\| nθ+ φ_n(θ) \| \leq ψ_n$ has infinitely many solutions. The main novelty is that the magnitude of the perturbation $|φ_n(θ)|$ is allowed to exceed $ψ_n$, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main resul…
We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq 2^{2^{n-1}}. By the conjecture, if a Diophantine equation has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of the degree and the coefficients of the equation. The conjecture implies that the set of Diophantine equations which have infinitely many solutions in integers (…
Hausdorff measures of sets in Exact Diophantine approximation
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；School of Statistics and Mathematics, Hubei University of Economics（湖北经济学院统计与数学学院）；Hubei Center for Data and Analysis（湖北省数据分析中心）
In this paper, we study the diophantine equation ${{σ}_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are $n={{F}_{2k+1}}{{F}_{2k+2m+1}}$, where both ${{F}_{2k+1}}$ and ${{F}_{2k+2m+1}}$ are Fibonacci primes. Meanwhile, we show that the twin primes conjecture holds if and only if the equation ${{σ}_{2}}(n)-{{n}^{2}}=2n+5$ has infinitely many solutions.
A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may think of this as a higher-dimensional version of the Gauss Class Number problem. Before now, no one has ruled out quaternion algebras over $\mathbf{Q}$ not already ruled out by the Albert classification. We rule out infinitely many such quaternion algebras by showing that for infinitely many $D$, the Atkin-Lehner quotient Shimura curve $X^D/w_D$ has no $\ma…
Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $P_m$ and $G_m/P_m$ have infinite rank for all $m>1$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.
Let $\mathcal{E}_{f}:y^2=x^3+f(t)x$, where $f\in\Q[t]\setminus\Q$, and let us assume that $\op{deg}f\leq 4$. In this paper we prove that if $\op{deg}f\leq 3$, then there exists a rational base change $t\mapstoϕ(t)$ such that on the surface $\cal{E}_{f\circϕ}$ there is a non-torsion section. A similar theorem is valid in case when $\op{deg}f=4$ and there exists $t_{0}\in\Q$ such that infinitely many rational points lie on the curve $E_{t_{0}}:y^2=x^3+f(t_{0})x$. In particular, we prove that if $\op{deg}f=4$ and $f$ is not an even polynomial, then there is a rational point on $\cal{E}_{f}$. Next…
