On additive decompositions of the set of primitive roots modulo p |
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Authors: | Cécile Dartyge András Sárközy |
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Institution: | 1. Institut élie Cartan, Université Henri Poincaré - Nancy 1, BP 239, 54506, Vand?uvre Cedex, France 2. Department of Algebra and Number Theory, E?tv?s Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary
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Abstract: | It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p 1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ . |
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