Number — page 5
A small quotient for every denominator
Pick a denominator, any whole number. Among the fractions with that denominator, is there always one whose continued fraction has no large entries — whose path down the Stern–Brocot tree never turns the same way more than five times running? Stanisław Zaremba guessed so in 1972, for a reason to do with sampling a square evenly. Every denominator up to a hundred thousand obeys, with only 6, 54 and 150 needing a five. The guess is proved for almost every denominator and not for all, and the reason it ought to be true is a dimension: the reals whose quotients stay at most five fill a set of dimension 0.837, enough to give each denominator thousands of good fractions on average, and the open question is whether averages are kept everywhere.
How close two sums of square roots come
Two different sums of square roots of whole numbers are never equal: square roots of different squarefree numbers are independent, so no cancellation can make them agree. But they can come very close. Among sums of two square roots of numbers up to 320, √88 + √289 and √171 + √177 agree to eight decimal places. How many digits must be computed, in the worst case, to tell two such sums apart is not known — the best proof allows a number of digits doubly exponential in the length of the sums, the examples suggest only a modest polynomial, and the gap between them is why nobody knows whether the shortest route through points in the plane can be checked quickly.