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On a Theorem of Sylvester and Schur

Published online by Cambridge University Press:  20 November 2018

D. Hanson
D. Hanson*
Affiliation:
University of Saskatchewan, Regina, Saskatchewan
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In 1892, Sylvester [7] proved that in the set of integers n, n+l,…, n+k—1, n> k > 1, there is a number containing a prime divisor greater than k. This theorem was rediscovered, in 1929, by Schur [6]. More recent results include an elementary proof by Erdös [1] and a proof of the following theorem by Faulkner [2]: Let pk be the least prime ≥2k; if n≥pk then has a prime divisor ≥pk with the exceptions and In that paper the author uses some deep results of Rosser and Schoenfeld [5] on the distribution of primes.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973 , pp. 195 - 199
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Erdös, P., A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934), 282–288.Google Scholar
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3. Hanson, D., On the product of the primes, Canad. Math. Bull., (1) 15 (1972), 33–37.Google Scholar
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5. Rosser, J.B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.Google Scholar
6. Schur, I., Einige Satze uber Primzahlen mit wendung auf Irreduzibilitatsfragen, Sitzungberichte der preussichen Akedemie der Wissenschaften, Phys. Math. Klasse, 23 (1929), 1–24.Google Scholar
7. Sylvester, J.J., On arithmetical series, Messenger of Mathematics, XXI (1892), 1–19, 87–120, and Mathematical Papers, 4 (1912), 687–731.Google Scholar