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. Erdős : Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged) 5 (1930–32), 194–198.
R. L. Graham, D. E. Knuth & O. Patashnik: Concrete Mathematics. A Foundation for 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.
P. Ribenboim: The New Book of Prime Number Records, Springer-Verlag, New York 1989.
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© 2004 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (2004). Bertrand’s postulate. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05412-3_2
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DOI: https://doi.org/10.1007/978-3-662-05412-3_2
Publisher Name: Springer, Berlin, Heidelberg
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