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Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, [2] it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip ("drawer principle" or "shelf principle").
where the last step follows by the pigeonhole principle and the sub-additivity of the probability measure. Each of the probabilities on the right-hand side converge to zero as n → ∞ by definition of the convergence of {X n} and {Y n} in probability to X and Y respectively.
Using the Pigeonhole Principle Thue, and later Siegel, managed to prove the existence of auxiliary functions which, for example, took the value zero at many different points, or took high order zeros at a smaller collection of points.
The existence of these polynomials was proven by Axel Thue; [1] Thue's proof used what would be translated from German as Dirichlet's Drawers principle, which is widely known as the Pigeonhole principle. Carl Ludwig Siegel published his lemma in 1929. [2] It is a pure existence theorem for a system of linear equations.
He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's Last Theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. [3]
This theorem is a consequence of the pigeonhole principle. Peter Gustav Lejeune Dirichlet who proved the result used the same principle in other contexts (for example, the Pell equation) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later. [2] The method extends to simultaneous ...
The pigeonhole principle says that at least three of them must be of the same colour; for if there are less than three of one colour, say red, then there are at least three that are blue. Let A , B , C be the other ends of these three edges, all of the same colour, say blue.
In cryptography, collision resistance is a property of cryptographic hash functions: a hash function H is collision-resistant if it is hard to find two inputs that hash to the same output; that is, two inputs a and b where a ≠ b but H(a) = H(b).