Speculations about certain outstanding properties of numbers
Euler writes that "without a doubt, the number of all the different fractions between the endpoints 0 and 1 is infinite; and since the number of all the integers is also infinite, it is manfiest that the multitude of all the ordinary fractions up to infinity is greater; and at the same time, there must be innumerably many different fractions between any two numbers that differ by one." He goes on to define πD as the number of integers less than D and relatively prime to D. He provides a table of πD up to D=100, then makes a table listing the number of fractions with denominators less than or equal to n, for n = 10, 20, …, 100, and poses the problem of finding this number of fractions for any given number N. The solution, of course, involves πN.
Original Source Citation
Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1780: II, pp. 18-30.
Opera Omnia Citation
Series 1, Volume 4, pp.105-115.