Abstract
The present article describes Erdős’s work contained in the following papers.
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[E1]
On the coefficients of the cyclotomic polynomials, Bull. Amer. Math. Soc. 52 (1946), 179–183.
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[E2]
On the coefficients of the cyclotomic polynomial, Portug. Math. 8 (1949), 63–71.
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[E3]
On the number of terms of the square of a polynomial, Nieuw Archief voor Wiskunde (1949), 63–65.
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[E4]
On the greatest prime factor of \(\mathop{\prod }\limits _{k=1}^{x}f(k)\), J. London Math. Soc. 27 (1952), 379–384.
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[E5]
On the sum \(\sum \limits _{k=1}^{x}d(f(k))\), ibid. 7–15.
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[E6]
Arithmetical properties of polynomials, ibid. 28 (1953), 436–425.
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[E7]
Über arithmetische Eigenschaften der Substitutionswerte eines Polynoms für ganzzahlige Werte des Arguments, Revue Math. Pures et Appl. 1 (1956) No. 3, 189–194.
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[E8]
On the growth of the cyclotomic polynomial in the interval (0,1), Proc. Glasgow Math. Assoc. 3 (1957), 102–104.
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[E9]
On the product \(\mathop{\prod }\limits _{k=1}^{n}(1 {-\zeta }^{a_{i}})\), Publ. Inst. Math. Beograd 13 (1959), 29–34 (with G. Szekeres).
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[E10]
Bounds for the r-th coefficients of cyclotomic polynomials, J. London Math. Soc. (2) 8 (1974), 393–400 (with R. C. Vaughan).
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[E11]
Prime polynomial sequences, ibid. (2) 14 (1976), 559–562 (with S. D. Cohen, M. B. Nathanson).
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[E12]
On the greatest prime factor of \(\mathop{\prod }\limits _{k=1}^{x}f(k)\), Acta Arith. 55 (1990), 191–200 (with A. Schinzel).
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Bibliography
F. V. Atkinson, On a problem of Erdős and Szekeres, Canad. Math. Bull. 4 (1961), 7–12.
D. Coppersmith and J. Davenport, Polynomials whose powers are sparse, Acta Arith. 58 (1991), 79–87.
C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press 1976.
H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 65 (1993), 227–235.
M. Nair, Power free values of polynomials, Mathematika 26 (1976), 159–183.
M. Nair, Multiplicative functions of polynomial values in short intervals, Acta Arith. 62 (1992), 257–269.
A. Schinzel, On the number of terms of a power of polynomial, ibid, 49 (1987), 55–70.
A. Schinzel and U. Zannier, Distribution of solutions of diophantine equations \(f_{1}(x_{1})f_{2}(x_{2}) = f_{3}(x_{3})\), where f i are polynomials, Rend. Sem. Mat. Univ. Padova 87 (1992), 39–68.
A. Schinzel and U. Zannier, On the number of non-zero coefficients of a power of a polynomial, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 20 (2009), 95–98.
G. Tenenbaum, Sur une question d’Erdős et Schinzel II, Inv. Math. 99 (1990), 215–224.
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289–295.
W. Verdenius, On the number of terms of the square and the cube of polynomials, Indag. Math. 11 (1949), 596–565.
D. Wolke, Multiplikative Funktionen auf schnell wachsenden Folgen, J. Reine Angew. Math. 251 (1971), 54–67.
U. Zannier, On composite lacunary polynomials and the proof of a conjecture of Schinzel, Invent. Math. 174 (2008), 127–138.
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Schinzel, A. (2013). Arithmetical Properties of Polynomials. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_17
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