A lower bound for linear forms in logarithms

Linear Independence Measures for Logarithms of Algebraic Numbers

“…It turns out that such a long extrapolation procedure has a cost: there is another factor log V n−1 (assuming V 1 ≤ · · · ≤ V n−1 ≤ V n ) in the final estimate. This is exactly the main result in [31], which improves earlier results by Baker in [3]. See also [20], I, as well as [37] for the p-adic case.…”

“…If one wishes to keep the conclusion |Λ| ≥ e −U then one should replace everywhere else U by U/2 n . Indeed this is just the choice of parameters in [31]. At the end of the n extrapolation steps one has S (n) 0 = S 0 /2 and S (n) 1 = 2 n S 1 .…”

On the nonexistence of $2$-cycles for the $3x+1$ problem

“…The nonexistence of nontrivial 2-cycles can alternatively be proved by applying a result of de Weger [9, p. 108]. He uses a result of Waldschmidt [8] to derive upper bounds for linear forms of the type a log 2 − b log 3. In particular he (implicitly) proves that the equation 1 < 2 k+ /3 k < 1 + 3 −0.1k has for k ≥ 32 no solutions.…”

Solving a Specific Thue-Mahler Equation

“…We now return to equation (1), or rather to (5). Consider the three conjugate relations a'(ß) = x -ya'(û).…”

“…For a £ K and h £ {0, 1,2} we will sometimes write ah instead of a (a). With this convention it suffices to solve (5) )ff = x-yI3 = ±/07r'1|^2öQl^2, where n0 £ {0, 1} and h £ {0, 1,2}, and where the variables are x, y e nx,n2£ Z>0 , and ax, a2 £ Z.…”