Lehmer问题的两个推广

Two Generalizations on a Problem of Lehmer

设素数P>2,整数C与P互素．对任意整数1≤a≤P-1,存在惟一的整数 1≤b≤P-1满足ab≡c mod P．Lehmer建议我们研究a与b的奇偶性不同的情形．本文给出了这一问题的两个推广,并获得了两个有趣的混合均值公式．

Let p 〉 2 be a prime, c be an integer with （c,p） = 1. For each integer a with 1 〈 a ≤ p- 1, there exists one and only one b with 1 ≤ b ≤ p- 1 such that ab ≡ c mod p. Professor Lehmer asked us to find the number of cases in which a and b are of opposite parity. In this paper, we give two generalizations on this problem, and obtain two interesting hybrid mean value formulae.