APPROXIMATING NUMBERS WITH MISSING DIGITS BY ALGEBRAIC NUMBERS

Perfect Numbers with Identical Digits

Suppose g ≥ 2. A natural number N is called a repdigit in base g if it has the shape a g −1 g−1 for some 1 ≤ a < g, i.e., if all of its digits in its base g expansion are equal. The number N is called perfect if σ(N) = 2N , where σ(N) := � d|N d is the usual sum of divisors function. We show that in each base g, there are at most finitely many repdigit perfect numbers, and the set of all such n...

On the Approximation to Algebraic Numbers by Algebraic Numbers

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...

Approximating Rational Numbers by Fractions

In this paper we show a polynomial-time algorithm to find the best rational approximation of a given rational number within a given interval. As a special case, we show how to find the best rational number that after evaluating and rounding exactly matches the input number. In both results, “best” means “having the smallest possible denominator”.

Algebraic Numbers By Barry Mazur

The roots of our subject go back to ancient Greece while its branches touch almost all aspects of contemporary mathematics. In 1801 the Disquisitiones Arithmeticae of Carl Friedrich Gauss was first published, a “founding treatise,” if ever there was one, for the modern attitude towards number theory. Many of the still unachieved aims of current research can be seen, at least in embryonic form, ...

On approximation of real numbers by real algebraic numbers