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A CLASS OF IRREDUCIBLE POLYNOMIALS ASSOCIATED WITH PRIME DIVISORS OF VALUES OF CYCLOTOMIC POLYNOMIALS

  • Michael Filaseta (a1)

Abstract

We prove that for every sufficiently large integer $n$ , the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$ . Similar results following from the same techniques are discussed.

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1. Apostol, T. M., Resultants of cyclotomic polynomials. Proc. Amer. Math. Soc. 24 1970, 462.10.1090/S0002-9939-1970-0251010-X
2. Dickson, L. E., History of the Theory of Numbers, Vol. I: Divisibility and Primality, Chelsea (New York, 1966).
3. Dresden, G., Resultants of cyclotomic polynomials. Rocky Mountain J. Math. 42 2012, 14611469.10.1216/RMJ-2012-42-5-1461
4. Dumas, G., Sur quelques cas d’irréductibilité des polynômes à coefficients rationnels. J. Math. Pures Appl. (9) 2 1906, 191258.
5. Filaseta, M., Coverings of the integers associated with an irreducibility theorem of A. Schinzel. In Number Theory for the Millennium, II (Urbana, IL, 2000), A. K. Peters (Natick, MA, 2002), 124.
6. Halberstam, H. and Richert, H.-E., Sieve Methods, Dover (Mineola, NY, 2011). Originally published by Academic Press (London Mathematical Society Monographs 4 (1974)).
7. Stewart, C. L., On divisors of Lucas and Lehmer numbers. Acta Math. 211 2013, 291314.10.1007/s11511-013-0105-y
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A CLASS OF IRREDUCIBLE POLYNOMIALS ASSOCIATED WITH PRIME DIVISORS OF VALUES OF CYCLOTOMIC POLYNOMIALS

  • Michael Filaseta (a1)

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