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We consider two cases: $n$ is odd or $n$ is even. If $n=2k+1$ then put $$ x=uz, \quad y=vz^{k}, $$ where $u, v$ are non-zero rational parameters. Thus $v^2z^{2k}=4u^{2k+1}z^{2k+1}+z^{2k}$ and after di …

Our question is a very special case of the general problem of restricted simultaneous Diophantine approximation. Thus, I believe that the best place to start is to consult the paper of Schmidt: Two qu …

There are only finitely many such matrices. Let us consder the circular matrix say $A$, with the first row $(i,j,k,m)$. Then $det(A)=FGH$, where $$ F=i-j+k-m,\; G=i+j+k+m,\;H=(i-k)^2+(j-m)^2 $$ We nee …

In a joint paper with Andrew Bremner (https://www.sciencedirect.com/science/article/pii/S0022314X13002527#se0070) we obtained several result concerning exitence of rational and integral points on surf …

Let $f\in\mathbb{Z}[x,y]$ be homogenous form of degree $\geq 3$. Then, the Diophantine equation of the form $F(x,y)=ap_{1}^{\alpha_{1}}\cdots p_{k}^{\alpha_{k}}$, where $a$ is a given non-zero integer …

As was observed by @JoshuaZ, for given $a, c\in\mathbb{N}$, the equation $(*)\; a^xy+x=c$ has only finitely many solutions. On the other hand, I show that for any $a$ and any $N\in\mathbb{N}$ one can …

Let $F(p,q,r,s)=p^6-q^6+r^3-s^3$. I prove that there are polynomials $p, q, r, s\in\mathbb{Q}[t]$ such that $deg_{t}(F(p(t),q(t),r(t),s(t)))=1$. This is enough to get the result. Indeed, if $F(p(t),q( …

To the books mentioned above I would add one more: Rational and Nearly Rational Varieties (Cambridge Studies in Advanced Mathematics) by J. Kollár, K. E. Smith, and A. Corti. The authors present a m …

The mentioned equation has infnitely many solutions. See the paper by A. Bremner and myself: A. Bremner, M. Ulas, On $x^a\pm y^b \pm z^c \pm w^d = 0, 1/a + 1/b + 1/c + 1/d = 1$, Int. J. Number Theory, …

The simplest answer is the following: for any integer $a\neq 12$ take $D=a^3-1728$. Then, the point $P=(a,1)$ is a point with integer coordinates on your curve. Moreover, the point $P$ is of infinite …

According to the conjecture of Schinzel and Tijdeman (Schinzel, Α., Tijdeman, R., On the equation $y^n = P(x)$, Acta Arith. 31 (1976), 199-204), if a polynomial $P(x)$ with rational coefficients has a …

The Conjecture 1 is true. We are looking for integers $m, n$ such that for sufficiently large prime $p$ we have $$ x_{1}^2\equiv 2mn\pmod{p},\quad x_{2}^2\equiv m^2-n^2\pmod{p}, \quad x_{1}^2\equiv m^ …

I think that the good place to start is the paper "Linear forms in logarithms" written by Sanda Bujačić and Alan Filipin, which is a part of Diophantine analysis course notes edited by J. Steuding (ht …

In order to get solutions of the congruence you are interested in let us consider the equation $x^2+y^2-x-Nz=0$. Using the trivial solution $x=1, y=0, z=0$, we parametrize all rational solutions by ta …