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Representation of integers by cyclotomic binary forms. (English) Zbl 1417.11028

For \(n \geq 1\), the cyclotomic binary form \(\Phi_n(X,Y)\) is defined by \(\Phi_n(X,Y)=Y^{\varphi(n)}\phi_n(X/Y)\), where \(\phi_n(X)\) is the cyclotomic polynomial of index \(n\) and degree \(\varphi(n)\) (Euler’s totient function).
The present paper proves that for each positive integer \(m\), the set \[ \{(n,x,y) \in \mathbb{N} \times \mathbb{Z}^2 | n \geq 3, \max \{|x|,|y|\}\geq 2, \Phi_n(x,y)=m \} \] is finite. Let \(a_m\) denote the cardinality of the above set. When \(a_m \neq 0\), \(m\) is said to be representable by a cyclotomic binary form, and \(a_m\) is then the number of such representations of \(m\). More precisely, it is shown that \(\varphi(n) \leq \frac {2}{\log 3} \log m\) and \(\max \{|x|,|y|\} \leq \frac {2}{\sqrt{3}} m^{1/\varphi(n)}\). This is a refinement of K. Győry’s result [Publ. Math. 24, 363–375 (1977; Zbl 0389.10018)], (see also [T. Nagell, Ark. Mat. 5, 153–192 (1964; Zbl 0119.27602)] for a slightly weaker result).
For \(N \geq 1\), the set of values taken by the forms \(\Phi_n\) for \(n \geq 3\) is defined by: \[ \mathcal{A}(\Phi_{\{n \geq 3\}} ; N) := \bigcup_ {n \geq 3} \mathcal{A}( \Phi_n ; N), \] where \[ \mathcal{A}(\Phi_n ; N) := \{m \in \mathbb{N} | m \leq N, m=\Phi_n(x,y) \, \text{for some} \, (x,y) \in \mathbb{Z}^2 \, \text{with} \, \max \{|x|,|y|\}\geq 2\} \] A description of the asymptotic cardinality of \(\mathcal{A}(\Phi_{\{n \geq 3\}} ; N)\) is given using the Selberg-Delange method. This implies that the set of integers \(m\) such that \(a_m \neq 0\) has natural density \(0\). The authors also deduce that if \(A_N=|\mathcal{A}(\Phi_{\{n \geq 3\}} ; N)|\) and \(M_N= \frac {1}{A_N}(a_1+\dots +a_N)\), then there exists a positive absolute constant \(\kappa\) such that \(M_N \sim \kappa \sqrt{\log N}\).
Numerical computations are also obtained [N. J. A. Sloane, The on-line encyclopedia of integer sequences. https://oeis.org/OEIS, A296095 and A299214].

MSC:

11E76 Forms of degree higher than two
12E10 Special polynomials in general fields

Software:

OEIS; PARI/GP; Maple
PDFBibTeX XMLCite
Full Text: DOI arXiv

Online Encyclopedia of Integer Sequences:

Numbers that are the sum of 2 nonzero squares.
Numbers of the form x^2 + xy + y^2, where x and y are positive integers.
Decimal expansion of Landau-Ramanujan constant.
Decimal expansion of Pi/sqrt(27).
a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...
Decimal expansion of log(2+sqrt(3))/sqrt(3).
Prime numbers of the form Phi_k(m), where k > 2, |m| > 1, and Phi_k(m) is the k-th cyclotomic polynomial evaluated at m.
Numbers of the form Phi_k(m) with k > 2 and |m| > 1.
Integers not represented by cyclotomic binary forms.
Integers represented by cyclotomic binary forms.
Number of representations of integers by cyclotomic binary forms.
Integers primitively represented by cyclotomic binary forms.
Prime numbers represented in more than one way by cyclotomic binary forms f(x,y) with x and y prime numbers and y < x.
Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.
Prime numbers represented by a cyclotomic binary form f(x, y) with x and y prime numbers and 0 < y < x.
Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.
Integers represented in more than one way by a cyclotomic binary form f(x,y) where x and y are prime numbers and 0 < y < x.
Integers represented by a cyclotomic binary form Phi{k}(x,y) with positive integers x and y where max(x, y) >= 2 and the index k is not prime.
Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.
a(n) = max{ p prime | n = Sum_{j in 0:p-1} x^j*y^(p-j-1)} where x and y are positive integers with max(x, y) >= 2 or 0 if no such representation exists.
Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.
Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.
Primes represented by cyclotomic binary forms.
Primes not representable by cyclotomic binary forms.
Primes represented by non-quadratic cyclotomic binary forms.
Decimal expansion of 1 / Product_{primes p == 5, 7, 11 (mod 12)} 1/(1 - 1/p^2).
Decimal expansion of Product_{primes p == 5, 7, 11 (mod 12)} 1/(1 - 1/p^2).
Positive integers representable by the two cyclotomic binary forms Phi_5(x,y) and Phi_12(u,v).

References:

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[4] Maple software, Univ. of Waterloo, Waterloo, ON. [M-W] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider’s method, III, Ann. Fac. Sci. Toulouse Math. (5), 1989, suppl., 43-75. [N1]T. Nagell, Contributions ‘a la th´eorie des corps et des polynˆomes cyclotomiques, Ark. Mat. 5 (1963), 153-192. [N2]T. Nagell, Sur les repr´esentations de l’unit´e par les formes binaires biquadratiques du premier rang, Ark. Mat. 5 (1965), 477-521. [OEIS] N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, https://oeis.org/.
[5] Pari/GP software, Univ. Bordeaux I, https://pari.math.u-bordeaux.fr/. [S-X]C. L. Stewart and S. Y. Xiao, On the representation of integers by binary forms, arXiv:1605.03427 (2016).
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