On decimal and continued fraction expansions of a real number

0. Introduction. Let x be an irrational number. We deal with the problem of finding from the decimal expansion of x, the first k (where k is a given integer) partial quotients of the regular continued fraction expansion of x. More precisely, for each n ≥ 1, denote by xn, yn with xn < x < yn the two consecutive nth decimal approximations of x. We assume that the integer n is such that the numbers xn and yn have finite continued fraction expansions which coincide up to order k, i.e., xn = [α0;α1, . . . , αk, . . .] and yn = [α0;α1, . . . , αk, . . .] for some integers αi. Since the set of numbers which have a continued fraction which begins with α0, . . . , αk is an interval, it follows that x = [α0;α1, . . . , αk, . . .], in other words α0, α1, . . . , αk are precisely the first k partial quotients of x. Writing the two rationals xn, yn as a quotient p/q of two integers, i.e., writing