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On a problem of D. H. Lehmer and its generalization

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On a problem of D. H. Lehmer and its generalization COMPOSITIO MATHEMATICA ZHANGWENPENG On a problem of D.H. Lehmer and its generalization Compositio Mathematica, tome 86, no 3 (1993), p. 307-316. <http://www.numdam.org/item?id=CM_1993__86_3_307_0> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 307 On a problem of D. H. Lehmer and its generalization* ZHANG WENPENG Department of Mathematics, Northwest University, Xi’an, China Compositio Mathematica (Ç) 1993 Kluwer Academic Publishers. Printed in the Netherlands. Received 6 January 1992; accepted 4 June 1992 Keywords: Congruence equation; D. H. Lehmer problem; Asymptotic formula; Prime power. Abstract. Let q &#x3E; 2 be an odd number. For each integer x with 0 x q and (q, x) = 1, we define x by xx - 1 (mod q) and 0 x q. Let r(q) be the number of integers x with 0 x q for which x and x are of opposite parity. The main purpose of this paper is to give an asymptotic formula for r(q) for odd numbers q of certain special types. 1. Introduction For any odd number n &#x3E; 2 and integer 0 x n with (n, x) = 1, we know that there exists one and only one x with 0 x n such that xx - 1 (mod n). Let r(n) be the number of cases in which x and x are of opposite parity. E.g., for n = 13, (x, x) = (1, 1), (2, 7), (3, 9), (4, 10), (5, 8), (6, 11), (12,12) so r(13) = 6. For n = p a prime, D. H. Lehmer [1] asks us to find r(p) or at least to say something nontrivial about. It is known that r(p) - 2 or 0 (m

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