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Here is parametrization involving rational numbers for the family $x^{2n+1}+a x^{2n}+b$. The discriminant factors as $\pm b^{2k}b(Ca^m+Db)$ for integers $k,m,C,D$. To make is square, consider $\pm b( …

Partial answer. Let $p_2(x,y,z)=1-xyz$ and $p_1(x,y,z)=x+y+z$. You want $p_2/p_1$ to be integer for integers $x,y,z$. Let $X=y^2 z + yz^2 - y - z + 1$. $p_2(X,y,z)/p_1(X,y,z)=1-y z$, so $x$ exists …

This will follow with very small $k$ from Schinzel's hypothesis H or as @Gerry Myerson points out Dickson's conjecture. Let $p$ be of the form $12n+1$. There are no congruence obstructions $p$ and $p …

Yes, this follows from Elkies' question here: $x^4+y^4$ powerful for relatively prime $x,y$ Let $u=427511122^2,v= 1322049209^2$. In the primitive triple $(u^2-v^2,2uv,u^2+v^2)$ the powerful sides ar …

Let $F \in \mathbb{Q}[x,y]$ be homogeneous of degree $d$ and $m$ is rational. Assume the curve $C : F(x,y)=m$ is irreducible and of genus greater than one. Currently, how many rational points $C$ ca …

Are all integers not congruent to 6 modulo 9 of the form $x^3+y^3+3z^2$ for possibly negative integers $x,y,z$? We have the identity $ (-t)^3+(t-1)^3+3 t^2=3t-1$. The only congruence obstruction we fo …

Finiteness of solutions follows from a generalization of the $abc$ conjecture, the $n$ conjecture for $\mathbb{Z}$: http://cr.yp.to/bib/1994/browkin.pdf Basically if $a_1 \ldots a_n$ are coprime inte …

I don't think simple characterizations exists, since there is an easy construction with all but one coefficients arbitrary. Set $y=1$ and $x,f_0 \ldots f_{d-1}$ to whatever you like (distinct is allo …

In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$ $$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C \prod_{ …

I think you need $d>2$. The answer to the first question is "no". For $F_n$ the Fibonacci numbers, $x,y=F_{2n},F_{2n+1}$ satisfy $x^2+xy-y^2+1=0$ Take the degree $3$ Thue equation $f(x,y)=x(x^2+xy- …

Let $g(x,y,z,t)=(x+y+z+t)^4-a h(x,y)$ where $h(x,y) \in \{x^4,xy^3,x^2y^2\}$ and $a$ is integer. Does $g$ represent infinitely many powers $r^n$ with $n > 4$, $x+y+z+t,ah(x,y)$ take distinct val …

The n-conjecture is a generalization of abc and basically says that the if $a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$ are coprime, then the radical of $a_1\cdots a_n$ can't be too sma …

Likely a mistake, but got very large exceptional set in Vojta's more general abc conjecture. In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprim …

I believe no polynomial in $\log_2 n$ algorithm is known, since this is related to open problems. Call your function $f(n)$. It is closely related to Chebyshev theta function $$ \vartheta(x)=\su …

Such curves with few monomials exist. So far they don't lead to sufficiently good triples/tuples because the large gcd ruins the quality. For positive even $k$ up to $40$ all of these are genus $1$ …