Linear forms in the logarithms of algebraic numbers

Gauss’ class number problem for imaginary quadratic fields

“…If £ is an elliptic curve over Q which is parametrised by modular functions, and K is a complex quadratic field such that the Mordell-Weil group E(K) of ^-rational points of E has odd rank, then the "canonical" X-rational point of E which is given by Heegner's construction has Tate height measured by L E/K (1).…”

“…For every e > 0 there exists an effectively computable constant c > 0 such that h(D) > cOoglDI) 1 [32] (that h(D) # 3 for 907 < -D < 10 2500 ), this gives the complete list of all imaginary quadratic fields with class number three. The complete list of all imaginary quadratic fields with class number 1, 2, or 4 would determine the complete finite Hst of all integers n which have a unique representation as a sum of three squares:…”

On a family of Thue equations of degree 16

“…where thec (j) i with 1 ≤ i ≤ 5 and 1 ≤ j ≤ 8 are obtained by directly computing the coefficients of the norm in equation (3). Note that this linear system is overdetermined, but we have the three parameters ζ 1 , ζ 2 and ζ 3 to play with.…”

“…This method is based on Baker's theorems on linear forms in logarithms [1,3]. Baker's method was further developed by Tzanakis and de Weger [21] and by Bilu and Hanrot [6,7].…”

Diophantine Properties of Linear Recursive Sequences I

“…The first general lower bound for linear forms in logarithms of algebraic numbers was proved by A. Baker [Ba1]. Since them several improvements and refinements appeared in the literature.…”