Abstract
We have seen that the sequence of prime numbers 2, 3, 5, 7, ... is infinite. To see that the size of its gaps is not bounded, let N := 2 · 3 · 5 · ... · p denote the product of all prime numbers that are smaller than k + 2, and note that none of the k numbers
is prime, since for 2 ≤ i ≤ k + 1 we know that i has a prime factor that is smaller than k + 2, and this factor also divides N, and hence also N + i. With this recipe, we find, for example, for k = 10 that none of the ten numbers
is prime.
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References
P. Erdös: Beweis eines Satzes von Tschebvschef, Acta Sci. Math. (Szeged) 5 (1930–32), 194–198.
R. L. Graham, D. E. Knuth & O. Patashnik: Concrete Mathematics. A Foundation fin- Computer Science, Addison-Wesley, Reading MA 1989.
G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers, fifth edition, Oxford University Press 1979.
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© 1998 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (1998). Bertrand’s postulate. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22343-7_2
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DOI: https://doi.org/10.1007/978-3-662-22343-7_2
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