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In number theory, Artin's conjecture on primitive roots states that a given integer ''a'' which is neither a perfect square nor −1 is a primitive root modulo infinitely many primes ''p''. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof. The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. Although significant progress has been made, the conjecture is still unresolved as of May 2014. In fact, there is no single value of ''a'' for which Artin's conjecture is proved. ==Formulation== Let ''a'' be an integer which is not a perfect square and not −1. Write ''a'' = ''a''0''b''2 with ''a''0 square-free. Denote by ''S''(''a'') the set of prime numbers ''p'' such that ''a'' is a primitive root modulo ''p''. Then # ''S''(''a'') has a positive asymptotic density inside the set of primes. In particular, ''S''(''a'') is infinite. # Under the conditions that ''a'' is not a perfect power and that ''a''0 is not congruent to 1 modulo 4, this density is independent of ''a'' and equals Artin's constant which can be expressed as an infinite product #: . Similar conjectural product formulas 〔 〕 exist for the density when ''a'' does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of ''C''Artin. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Artin's conjecture on primitive roots」の詳細全文を読む スポンサード リンク
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