Particularly from the work of Carl Friedrich Gauss around 1800 there emerged a procedure to find solutions to any quadratic Diophantine equation in two variables—in effect by reduction to the Pell equation x 2  a y 2 + 1 (see page 944 ), and then computing ContinuedFraction[ √ a ] . … Starting in the late 1800s and continuing ever since a series of progressively more sophisticated geometric and algebraic views of Diophantine equations have developed. … But as one continues the enumeration there are increasingly a few equations that seem more and more difficult to handle.

The number of constraints which yield solutions of specified lengths Length[s] for r = 2 and r = 3 are as follows (the boxes at the end give the number of cases with no solution): With r = 2 , as n increases an exponentially decreasing fraction of possible constraints have solutions; with r = 3 it appears that a fraction more than 1/4 continue to do so.

. • 1700s: Leonhard Euler and others compute continued fraction representations for numbers with simple formulas (see pages 143 and 915 ), noting regularity in some cases, but making no comment in other cases. • 1700s and 1800s: The digits of π and other transcendental numbers are seen to exhibit apparent randomness (see page 136 ), but the idea of thinking about this randomness as coming from the process of calculation does not arise. • 1800s: The distribution of primes is studied extensively—but mostly its regularities, rather than its irregularities, are considered.