Given a nonzero integer n, Gupta and Saha [‘Integer solutions of the generalised polynomial Pell equations and their finiteness: the quadratic case’, Canad. Math. Bull., to appear] classified all polynomials $x^2+ax+b\in {\mathbb {Z}}[x]$ for which the polynomial Pell equation $P^2-(x^2+ax+b)Q^2=n$ has solutions ${P,Q\in {\mathbb {Z}}[x]}$ with $Q\neq 0$. We generalise their work to the equation $P^2-(f^2+af+b)Q^2=nR$, where f is a fixed polynomial in ${\mathbb {Z}}[x]$. As an application of our results, we study the equation $P^2-D(f)Q^2=n$, where D is a monic, quartic and non square-free polynomial in ${\mathbb {Z}}[x]$. This extends Theorem 1.4 of Scherr and Thompson [‘Quartic integral polynomial Pell equations’, J. Number Theory 259 (2024), 38–56].

The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^{2}-Du^{2}=1$ is less than $D^{1/2+\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$-analogue of van der Corput method to algebraic exponential sums with smooth moduli.

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed $l$, the number of pairs of squareful numbers $n$, $n+l$ such that $n$is less than a given quantity.

In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$ , say $d$ , one of these two polynomials is irreducible and another has an irreducible factor of degree $d$ , so that $\alpha =M\left( P \right)-bM\left( Q \right)$ with irreducible polynomials $P,Q\in \mathbb{Z}\left[ X \right]$ of degree $d$ and a positive integer $b$ . Finally, if $d\le 3$ , then one can take $b=1$ .