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Non-sums of two cubes of algebraic integers. (English) Zbl 1465.11082

This paper is related to the author’s earlier investigations. In [Albanian J. Math. 8, No. 2, 49–53 (2014; Zbl 1366.11065)], the author proved that given a number field \(K \subset \mathbb R\) and a positive integer \(a\), there exist many prime elements \(\pi\) of the ring \(\mathcal O_K\) of integers of \(K\) for which the equation \(a^2 = x^4-y^2\) does not have solutions in \(\mathcal O_K\).
In the present paper it is shown that if \(K\) is a number field, then there exists a set of rational prime numbers, of positive Dirichlet density, such that none of the numbers in the set can be written as a sum of two cubes of integers of \(K\).
The proof uses class field theory and bounds on the Carmichael function by P. Pollack [Colloq. Math. 147, No. 2, 217–220 (2017; Zbl 1422.11224)].

MSC:

11D25 Cubic and quartic Diophantine equations
11R04 Algebraic numbers; rings of algebraic integers
11N37 Asymptotic results on arithmetic functions
11R42 Zeta functions and \(L\)-functions of number fields
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References:

[1] N. Childress,Class Field Theory, Springer, 2009. · Zbl 1165.11001
[2] J. Milne,Algebraic Number Theory, www.jmilne.org/math/CourseNotes/ant.html (2011).
[3] J. Neukirch,Algebraic Number Theory, Springer, 1999. · Zbl 0956.11021
[4] P. Pollack,A simple proof of a theorem of Hajdu-Jarden-Narkiewicz, Colloq. Math. 147 (2017), 217-220. · Zbl 1422.11224
[5] A. Schinzel,Families of curves having each an integer point, Acta Arith. 40 (1982), 399-420. · Zbl 0494.10007
[6] C. L. Siegel,Sums ofmthpowers of algebraic integers, Ann. of Math. 46 (1945), 313-339. · Zbl 0063.07010
[7] M. Tsunekawa,Sum of two fourth powers of integers, Nagoya Math. J. 18 (1961), 53-61. · Zbl 0102.28103
[8] A. Zinevičius,On the congruent number problem over integers of real number fields, Albanian J. Math. 8 (2014), 49-53. · Zbl 1366.11065
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