On the Harborth constant of C 3 C 3p
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 613-633.

For a finite abelian group (G,+,0) the Harborth constant g(G) is the smallest integer k such that each squarefree sequence over G of length k, equivalently each subset of G of cardinality at least k, has a subsequence of length exp(G) whose sum is 0. In this paper, it is established that g(C 3 C 3p )=3p+3 for prime p3 and g(C 3 C 9 )=13.

Soit (G,+,0) un groupe abélien fini. La constante de Harborth de G, notée g(G), est le plus petit entier k tel que toute suite d’éléments deux à deux distincts de G de longueur k, de manière équivalente tout sous-ensemble de G de cardinal au moins k, admet une sous-suite de longueur exp(G) dont la somme soit 0. Dans cet article, il est démontré que g(C 3 C 3p )=3p+3 pour tout nombre premier p3 et que g(C 3 C 9 )=13.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1097
Classification: 11B30, 20K01
Keywords: finite abelian group, zero-sum problem, Harborth constant, squarefree sequence
Guillot, Philippe 1, 2; Marchan, Luz E. 3; Ordaz, Oscar 4; Schmid, Wolfgang A. 1, 2; Zerdoum, Hanane 1, 2

1 Laboratoire Analyse, Géométrie et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France
2 Laboratoire Analyse, Géométrie et Applications (LAGA, UMR 7539), COMUE Université Paris Lumières, Université Paris 8, CNRS, 93526 Saint-Denis cedex, France
3 Escuela Superior Politécnica del Litoral, ESPOL, Facultad de ciencias naturales y matemática. Campus Gustavo Galindo, km 30.5, vía Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador
4 Escuela de Matemáticas y Laboratorio MoST, Centro ISYS, Facultad de Ciencias, Universidad Central de Venezuela, Ap. 47567, Caracas 1041–A, Venezuela
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     title = {On the {Harborth} constant of $C_3 \oplus C_{3p}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Guillot, Philippe; Marchan, Luz E.; Ordaz, Oscar; Schmid, Wolfgang A.; Zerdoum, Hanane. On the Harborth constant of $C_3 \oplus C_{3p}$. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 613-633. doi : 10.5802/jtnb.1097. http://www.numdam.org/articles/10.5802/jtnb.1097/

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