On the number of solutions to systems of Pell equations
Two important results on Diophantine equations are proved in this paper. (1) If \(a>1\), \(b>1\) are distinct integers and \(c,d\) are positive integers, then the system of equations \[ ax^2-cz^2=1, \quad by^2-dz^2=1 \] has at most two solutions in positive integers \(x,y,z\). This result is best possible, since there exist systems as above (actually, infinitely many) for which the system has two solutions. (2) If \(a,b\) are positive integers, then the system of equations \[ x^2-ay^2=1,\quad z^2-bx^2 =1 \] has at most two solutions in positive integers \(x,y,z\). This result, most likely, is not best possible, in the sense that no such system is known with two solutions. The problem of determining an upper bound for the number of solutions to such systems and, more general, to systems with left-hand sides as above and right-hand sides integers not necessarily equal to 1, has attracted the interest of number-theorists for the last 20 years. Much earlier, due to Thue and Siegel it has been known that the number of solutions is finite. The Introduction of the paper contains quite a number of relevant references. Though quite technical, this paper is very well written, so that anybody seriously interested in applications of Baker's method to Diophantine equations will greatly profit from its study. Moreover, I would recommend this paper to graduate students whose research orientation is to classical Diophantine equations as an especially useful reading. A very rough sketch of the proof of the first result follows; the proof of the second result follows similar lines. It is proved first that the number of solutions to the system appearing in (1) is equal to that of a similar system with \(c=a-1\) and \(d=b-1\) (where, of course, \(a\) and \(b\) are, in general, distinct from the \(a\) and \(b\) of the initial system). Next, let \(\alpha=\sqrt{a}+\sqrt{a-1}\), \(\beta=\sqrt{b}+\sqrt{b-1}\). Any solution \(x,y,z\) implies the existence of positive integers \(j,k\) satisfying \[ {\alpha^j-\alpha^{-j}\over 2\sqrt{a-1}}={\beta^k-\beta^{-k}\over 2\sqrt{b-1}}; \] this common value is, actually, the value of \(z\). Obviously, \(j=k=1\) gives the solution corresponding to \(z=1\) and the problem is to show that there are no distinct pairs \((j_2,k_2)\), \((j_3,k_3)\) satisfying the condition above. If the condition above is satisfied then the authors consider \[ \Lambda =\log \left({b-1\over a-1}\right)+2j\log\alpha-2k\log\beta, \] for which an upper bound in terms of \(j\) can be easily calculated. If one assumes the existence of \((j_2,k_2)\), \((j_3,k_3)\) as above with \(j_3>j_2\) and \(k_3>k_2\), then a ``gap principle'' is proved, i.e. the larger pair is much larger than the smaller one; more precisely, it is proved that \(j_3>c_1j_2\beta^{c_2(k)}\), where the constants are made explicit in the paper (depending on various cases). Using a sharp lower bound for linear forms in three logarithms due to E. M.~Matveev, the authors obtain a lower bound for \(\log\Lambda\) when \(k=k_3\) and \(j=j_3\) in terms of \(\log k_3\) and, of course, of \(\log\alpha\) and \(\log\beta\). This, combined with the ``gap principle'' and the lower bound for \(\Lambda\) implies \(\max\{a,b\}<3.1\times 10^{51}\). In order to reduce this bound considerably, the authors write \({k_2-1\over 2}\Lambda=\log\alpha_2-k_3\log\alpha_1\), where \[ \alpha_2= \left({b-1\over a-1}\right)^{{k_2-1\over 2}}\alpha^{(k_2-1)j_3-(j_2-1)k_3}. \] This is now a linear form in \textit{two} logarithms. The algebraic numbers in the logarithms are considerably more complicated but, the advantage is that the authors can use a very sharp result on linear forms in two logarithms due to M.~Laurent, M.~Mignotte and Y.~Nesterenko, which permits them to reduce the previous bound for \(\max\{a,b\}\) to \(4\times 10^{38}\). Now I quote from the authors' introduction: ``To complete the proof, we use computers to perform two types of computations. On the one hand, standard techniques from computational Diophantine approximation yield the conclusion that small values of \(\max\{a,b\}\), say, smaller than 2000, are not compatible with the existence of three positive solutions [\dots]. On the other hand, various verifications eliminate the possibility that the third solution exists when \(\max\{a,b\}\) is in the domain excluded for other reasons.''
scientific article; zbMATH DE number 5190748
scientific article