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On a Theorem of Sylvester and Schur

  • D. Hanson (a1)
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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.

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References
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1. Erdös, P., A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934), 282288.
2. Faulkner, M., On a theorem of Sylvester and Schur, J. London Math. Soc. 41 (1966), 107110.
3. Hanson, D., On the product of the primes, Canad. Math. Bull., (1) 15 (1972), 3337.
4. Moser, L., Insolvability of , Canad. Math. Bull. (2) 6 (1963), 167169.
5. Rosser, J.B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 6494.
6. Schur, I., Einige Satze uber Primzahlen mit wendung auf Irreduzibilitatsfragen, Sitzungberichte der preussichen Akedemie der Wissenschaften, Phys. Math. Klasse, 23 (1929), 124.
7. Sylvester, J.J., On arithmetical series, Messenger of Mathematics, XXI (1892), 1–19, 87–120, and Mathematical Papers, 4 (1912), 687731.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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