We show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive number.

Soient $p$ et $q$ deux nombres premiers distincts et $X^{pq}/w_q$ le quotient de la courbe de Shimura de discriminant $pq$ par l’involution d’Atkin-Lehner $w_q$. Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si $p$ et $q$ satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si $p$ est assez grand par rapport à $q$, alors le quotient $X^{pq}/w_q$ n’a pas de point rationnel non spécial.

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL${}_2$ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL${}_3$ have been considered, in which analogous GL${}_3$-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL${}_3\left(\mathbb{Z}\right)$. We give formulas for the...

We consider Weil sums of binomials of the form $W_{F,d}\left(a\right)={\sum }_{x\in F}\psi (x^d-ax)$, where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{\times }\left|\right)=1$, and $a\in F^{\times }$. If we fix F and d, and examine the values of $W_{F,d}\left(a\right)$ as a runs through $F^{\times }$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{\times }|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n odd, and $d=3^r+2$ with...