In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire N\'eron--Severi group of this surface and find all low degree curves on it.

In the article under consideration (Canad. Math. Bull. \textbf{47} (2004), pp.~373--388), Lemma 6 is not true in the form presented there. Lemma 6 is used only in the proof of part (i) of Theorem 9. We note, however, that part (i) of Theorem 9 in question is a special case of a theorem by Bennet, Bruin, Gy\H{o}ry and Hajdu.

We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obl\'ath for the case of squares, and an extension of a theorem of Gy\H ory on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in $n,y$ when $b=d=1$. We show that there are only finitely many solutions in $n,d,b,y$ when $k\geq 3$, $l\geq 2$ are fixed and $k+l>6$.

Soit $K$ un corps de nombres de degr\'e sur $\mathbb{Q}$ inf\'erieur ou \'egal \`a $2$. On se propose dans ce travail de faire quelques remarques sur la question de l'existence de deux \'el\'ements non nuls $a$ et $b$ de $K$, et d'un entier $n\geq 4$, tels que l'\'equation $ax^n + by^n = 1$ poss\`ede au moins trois points distincts non triviaux. Cette \'etude se ram\`ene \`a la recherche de points rationnels sur $K$ d'une vari\'et\'e projective dans $\mathbb{P}^5$ de dimension $3$, ou d'une surface de $\mathbb{P}^3$.

The $abc$-conjecture is applied to various questions involving the number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary over integers $n$.

We combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles' deep machinery.