Sei Q(x, k, I) die Aazahl der quadratfreien natiirliehen Zahlen < x, welehe =/(rood k) sind (k sei eine natiirliche Zahl und 0 ~ 1 < k). Wenn der grSl]te gemeinsame Teller d = (k, l) nicht quadratfrei ist, so gibt es keine quadratfreien Zahlen, wetche -/(rood/c) sind. Landau I hat bewiesen, dab = 2(k, l) = lim Q(x, k, 1)/x > 0 X-~ oo 1909, Bd. 2, S. 633~636.

No abstract

Abstract. It is proved that the equation tan (k x/m) = k tan x/m has no solution in integers k and m with k >~ 2, m ~> 3. This answers a question concerning the problem of approximating a convex disc by polygons.1. The object of the present paper is to show that the equationm has no integral solutions k,m with k >~ 2 and m >~ 3. Equation (1) is closely connected with the following problem: Let K be a convex disc, i.e., a convex compact subset of the Euclidean plane with non-empty interior. Denote by p (K) the perimeter of K, and by pro(K) the maximum of the perimeters of all convex m-gons which are inscribed in K. SCHNEIDER [3] proved that m 7~m for any convex disc K and any integer m >/3 (cf. also FEJES TOTH [1], p. 191). IfKis a circle, equality occurs in (2) for every m/> 3. If(l) has no integral solution k >~ 2 for a given integer m >~ 3, then in (2) equality holds only for the circle. In [3] it is shown that (1) is unsolvable for m ~< 21 (m odd) and m ~< 42 (m even). In the other cases the question remained open and will be answered by the theorem below. Another problem led MEISSNER [2] to the more general equation 19 Monatshefte ffir Mathematik, Bd. 102/4

No abstract

No abstract