A note on the least totient of a residue class
نویسنده
چکیده
Let q denote a large prime number, a be any integer. Let N(q, a) denote the smallest positive integer n for which φ(n) ≡ a (mod q). The number N(q, a) exists. Indeed, if a+1 ≡ 0 (mod q) then one can take n = q. Otherwise, one can take n to be a prime from the arithmetical progression a + 1 (mod q). The problem of upper bound estimates for N(q, a) has been a subject of study of the work of Friedlander and Shparlinski [4]. In the present paper we obtain a new upper bound for N(q, a). Throughout, we use the notation A . B or B & A to indicate that A ≪ Bq for any fixed ε > 0, where the implied constant may depend on ε. Theorem 1. For any prime q and integer a, we have N(q, a) . q. Theorem 1 improves the bound N(q, a) . q of [4].
منابع مشابه
Least Totient in a Residue Class
For a given residue class a (mod m) with gcd(a,m) = 1, upper bounds are obtained on the smallest value of n with φ(n) ≡ a (mod m). Here, as usual φ(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler function in reduced residue classes modulo m. Some discussion and results are also given for classes...
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تاریخ انتشار 2008