By Édouard Lucas

This Elibron Classics publication is a facsimile reprint of a 1895 variation through Gauthier-Villars et fils, Paris.

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The final surprise is that, apart from these three cases, all of the other H n are the infinitely recurring variety. Our proof of this remarkable fact will take us from comparatively shallow to very deep water, with the need of a most profound and significant result of number theory: the Bertrand Conjecture. In 1845 the French mathematician Joseph Bertrand (18221900) conjectured that for every positive integer n > 1, there exists at least one prime p satisfying n < p < 2n (having verified it for n < 3 000 000).

Kempner, who in 1914 considered what would happen if all terms are removed from it which have a particular digit appearing in their denominators. For example, if we choose the digit 7, we would exclude the terms with denominators such as 7, 27, 173,33779, etc. There are 10 such series, each resulting from the removal of one of the digits 0, 1, 2, ... , 9, and the first question which naturally arises is just what percentage of the terms of the series are we removing by the process? For example, if we remove all terms involving 0 we are left with 1 1 1 1 1 1 1 + - + - + ...

Using his terminology, w is an 'infinitely smaIl' number and n an 'infinitely large' one, with I representing the logarithm. Since w is 'infinitely small', 1(1 + w) = wand therefore y = I (I + w)n = nw. Now let x = (I + w)n, then 1 + w = x I / n and w = x I / n - 1, which means that Ix = y = n(x I / n - 1). He then argued that there are n (complex) values of x I / n for any x and since n is an infinite number, there must be an infinite number of values of Ix. He continued by pointing out that all but one of the values would involve R, presaging one of the most subtle ideas of the next century's complex function theory, the Riemann surface.

### L'arithmétique amusante by Édouard Lucas

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