Polynomials with values which are powers of integers

Rachid Boumahdi; Jesse Larone

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 2, page 119-125
  • ISSN: 0044-8753

Abstract

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Let P be a polynomial with integral coefficients. Shapiro showed that if the values of P at infinitely many blocks of consecutive integers are of the form Q ( m ) , where Q is a polynomial with integral coefficients, then P ( x ) = Q ( R ( x ) ) for some polynomial R . In this paper, we show that if the values of P at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form m q where q is an integer greater than 1, then P ( x ) = ( R ( x ) ) q for some polynomial R ( x ) .

How to cite

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Boumahdi, Rachid, and Larone, Jesse. "Polynomials with values which are powers of integers." Archivum Mathematicum 054.2 (2018): 119-125. <http://eudml.org/doc/294767>.

@article{Boumahdi2018,
abstract = {Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q(m)$, where $Q$ is a polynomial with integral coefficients, then $P(x)=Q( R(x))$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P(x)=( R(x))^q$ for some polynomial $R(x)$.},
author = {Boumahdi, Rachid, Larone, Jesse},
journal = {Archivum Mathematicum},
keywords = {integer-valued polynomial},
language = {eng},
number = {2},
pages = {119-125},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Polynomials with values which are powers of integers},
url = {http://eudml.org/doc/294767},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Boumahdi, Rachid
AU - Larone, Jesse
TI - Polynomials with values which are powers of integers
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 2
SP - 119
EP - 125
AB - Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q(m)$, where $Q$ is a polynomial with integral coefficients, then $P(x)=Q( R(x))$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P(x)=( R(x))^q$ for some polynomial $R(x)$.
LA - eng
KW - integer-valued polynomial
UR - http://eudml.org/doc/294767
ER -

References

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  9. Shapiro, H.S., 10.2307/2310169, Amer. Math. Monthly 64 (1957). (1957) DOI10.2307/2310169
  10. Szalay, L., Superelliptic equations of the form y p = x k p + a k p - 1 x k p - 1 + ⋯ + a 0 , Bull. Greek Math. Soc. 46 (2002), 23–33. (2002) MR1924066
  11. Tijdeman, R., 10.4064/aa-29-2-197-209, Acta Arith. 29 (2) (1976), 197–209. (1976) DOI10.4064/aa-29-2-197-209
  12. Voutier, P.M., 10.1006/jnth.1995.1090, J. Number Theory 53 (1995), 247–271. (1995) DOI10.1006/jnth.1995.1090
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