The Pigeonhole Principle, Two Centuries Before Dirichlet

Abstract

The name of Dirichlet is commonly associated with the pigeonhole principle, because it is widely believed that he was the first to state it. In the writings of Dirichlet, the application of the principle is to be found in 1842 in [7] and [5] (both reproduced in [8], see pp. 579–580 and 633–638) and, later, in [6]. It seems that no one knows any precise previous reference, even if 1834 is frequently mentioned as the year of the discovery (for some details on later references, see [15]). In [7] and [5], the pigeonhole principle is used to prove complex and multidimensional versions of Equation 1. In the second reference, of 1863 (hence published 4 years after Dirichlet’s death), it is used to provide a proof of the existence of infinitely many integers x and y such that x yD\1 þ 2 ffiffiffiffi D p (for D integer and not a perfect square) which does not rely on continued fractions. In these publications Dirichlet does not assign any name to the principle, nor does he pretend that this principle is new. In a later work he calls it the ‘‘Schubfachprinzip.’’ Anyway, it appears that Dirichlet was not the first to make use of the principle that bears his name. The pigeonhole principle appeared no less than two centuries before him.