The following results related to the set of solutions of the Pell equation \[ X^2 - dY^2 = m \tag{\text{*}} \] are proved:
Theorem 1. Let \(X_1 < X_2 < X_3\) be the \(X\)-components of three positive solutions to (\(\ast\)), forming a geometric progression, i.e. they fulfil \(X_1 X_3 = X_2^2\). Then, \(X_3 < 1645683\,|m|^{20}\). Similarly, if \(Y_1 < Y_2 < Y_3\) are the \(Y\) - components of three positive solutions to (\(\ast\)) which form a geometric progression, then \(Y_3 < 1645683\,|m|^{20} /d\).
Theorem 2. Let \((X_i , Y_i )\), \(i = 1, 2, 3\), be three solutions to (*), with \(X_i Y_i = 0\) for every \(i\), which satisfy the linear equation \(aX_1 + bX_2 + cX_3 + f = 0\), where \(a, b, c, f\) are given integers with \(abc \neq 0\), and, in case that \(f = 0\), the positive integers \(|a|, |b|, |c|\) represent the sides of a (possibly degenerated) triangle. Then, either \(\max_{1 \leq i \leq 3} |X_i | \leq C\), where \(C\) is a constant depending on \(\max\{|a|, |b|, |c|\}, f\) and \(m\), or one of four exceptional cases occurs.
It is important to note that the constant \(C\) is explicitly stated in the paper; it is only for the sake of shortness of this review that we avoid stating the explicit value of \(C\). Similarly, the four exceptional cases are totally explicitly stated in the paper. Note that if, for example, \(X_1 < X_2 < X_3\) and \(a = c = 1, b = -2, f = 0\), then \(X_1 , X_2 , X_3\) are in arithmetic progression, and Theorem 2 furnishes an upper bound for non-constant positive arithmetic progressions. It is also interesting to note that Theorem 1 is a consequence of Theorem 2, but the proof is quite tricky.
From the Introduction (Section 1 of the paper): “In the next section [= Section 2] we will prove Theorem 1.3 [= Theorem 2 above], which is essential for proving Theorem 1.1 [= Theorem 1 above] in the subsequent Section 3. The proof of Theorem 1.3 is, beside the use of Gröbner bases, elementary. The cases of fixed three and five term geometric progressions is discussed in Section 4 and the case of fixed four term geometric progressions is treated in Section 5. The treatment of fixed five term geometric progressions makes use of Faltings’ theorem [...] on rational points of curves of genus \(> 1\) and our result is therefore non-effective. On the other hand, in the case of four term geometric progressions we are led to elliptic curves and we can effectively compute Pell equations that admit a given geometric four term progression. The last section is devoted to geometric progressions in Lucas sequences. The use of the primitive divisor Theorem due to Bilu, Hanrot and Voutier [...] breaks the problem down to some elementary considerations.” The result concerning Lucas sequences is as follows: Let \(u_n = (\alpha^n - \beta^n )/(\alpha - \beta)\) be the typical Lucas sequence, where \(\alpha= (a + \sqrt b)/2\) and \(\beta = (a -\sqrt b)/2\), with \(a, b\) non-zero integers, \(b \neq a^2\) and \(\alpha/\beta\) is not a root of unity.
Theorem 3. Assume that there are three distinct indices \(n, k, l\) such that \(u_k u_l = u\overset{2} {n}\). Then, except for the trivial case where \(u_k , u_l , u_n \in {\pm1}\) the only solutions are (\(u_1 , u_2 , u_4 ) = (u_3 , u_2 , u_4 ) = (1, -2, 4)\) with \(a = -2\) and \(b = -8\).