THE EQUATIONS 3x²−2 = y² AND 8x²−7 = z²

On a family of Thue equations of degree 16

“…Now we may state a variant of a result due to Baker and Davenport [4] Proof. We consider equation (20), divide it by log 1 and multiply it by q.…”

“…Utilizing the generalized method of Tzanakis and using lower bounds for linear forms in logarithms, we find a crude upper bound for η t (section 4). An application of a method due to Baker and Davenport (see [4] or section 5) shows t > 10 7 . In section 6 we use this new lower bound for t to sharpen our first bound for η t .…”

“…Using a method due to Baker and Davenport [4] (see also Section 5) or a method based on the LLL-algorithm (see [17, section VI.3]) we can reduce the usually huge first bound N 0 to a suitable smaller bound N , i.e. we can find suitable small upper bounds for |U 2 | and also |Y |.…”

“…In this section, we want to apply a method introduced by Baker and Davenport [4]. This method yields new upper bounds for n 2 for every specific t. So it is possible to show that the only solutions to (1) are trivial, provided t < 10 7 .…”

On the size of Diophantine m-tuples

“…Since ce + n 2 is a perfect square, we have that ce+n 2 ≥ 0. On the other hand, the assumption is that c > |n| 3 …”