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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].

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].

Reviewer: Lekbir Chakri (Meknès)

### Keywords:

cyclotomic binary forms; cyclotomic polynomials; Euler’s totient function; families of Diophantine equations; Thue Diophantine equations; representation of integers by binary forms
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\textit{É. Fouvry} et al., Acta Arith. 184, No. 1, 67--86 (2018; Zbl 1417.11028)

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[2] | L. K. Hua, Introduction to Number Theory , Springer, Berlin, 1982. [I-K]H. Iwaniec and E. Kowalski, Analytic Number Theory , Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, RI, 2004. |

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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/. |

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