Integer-Coefficient Polynomials Have Prime-Rich Images

• Autores: Bryan Bischof, Javier Gómez-Calderón, Andrew Perriello
• Localización: Mathematics magazine, ISSN 0025-570X, Vol. 83, Nº. 1, 2010, págs. 55-57
• Idioma: inglés
• DOI: 10.4169/002557010x480008
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• Resumen

  • It is well known that a nonconstant polynomial f (x) with integer coefficients produces, for integer values of x, at least one composite image. In this note, we use Taylor expansions to improve this elementary result, showing that f (x) takes an infinite number of composite values. Given a positive integer n, we show that f (x) takes an infinite number of values that are divisible by at least n distinct primes, and an infinite number of values that are divisible by pn for some prime p.