Exponential diophantine equations with four terms
This article gives some examples how to make exponential diophantine equations more practical. The authors take the large exponential bounds for solutions given by Baker's method to computational available bounds. Let \(p\) and \(q\) be distinct primes less than 200. The main theorems are: (1) Every solution of the equation \(p^ x q^ y\pm p^ z \pm q^ w \pm 1=0\) with \(zw>0\) satisfies \(\max(p^ x,q^ y,p^ z,q^ w)\leq 2^{15}\). (2) Every solution of the equation \(p^ x\pm q^ y\pm p^ z\pm q^ w=0\) subject to \(x\geq z\), \(y\geq w\), \(p^ x>q^ y\), \(xy>zw\) satisfies \(\max(p^ x,q^ y,p^ z,q^ w)\leq 2^{15}\). (3) Let \(n\) be a positive integer with \(n\leq 5000\). All solutions of \(p^ x-q^ y=n\) satisfy \(\max(p^ x,q^ y)\leq 29^ 4=707281\). The only solutions with \(p^ x>199^ 2\) are given by \(29^ 4-89^ 3=2312\), \(5^ 8-73^ 3=1608\), \(2^{17}-19^ 4=751\), \(17^ 4-43^ 3=4041\), \(43^ 3-5^ 7=1382\), \(41^ 3-2^{16}=3385\).
scientific article; zbMATH DE number 57231
scientific article