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 Ï D as the number of integers less than D and relatively prime to D. He provides a table of Ï D 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 Ï N.
Original Source Citation
Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1780: II, pp. 18-30.
Opera Omnia Citation
Series 1, Volume 4, pp.105-115.