Artigo Note on perfect and multiply perfect numbers Cópia em PNG Índice     | Índice > 21       Tipo:image/png Tamanho:48 KB Preto/branco:01 bpp Resolução:300 dpi NOTE ON PERFECT AND MULTIPLY PERKECT NUMBERS 21 To verify (3) we need only repeat G-UÚN'S proof. replacing 2 by s throughout. If n is even, pl = 2, and Ave have i~» t ^ — t* NOAV, />, ^ p2 f 2 i — 4 (è = l ,-•-,/), and só we obtain But, / p8 + 2 1 — 4 pa + 2 f — 5 and só from Avhich Ave can obtain (4). The left-hand sides of both (3) and (4) must be at least 3, só the right-hand sides must be at least 4. From this AVO obtain the COROLLARY. Let u be an (s — l)-fold multiply perfect number. Then, if n i* odd, t X s2 - l, and i n is even, The only other estiinate of this type Avhich AAre could find in the literature is the trivial one, £>(s —l)/2, Avhich is given by H.-J. KAXOLD in [2_ It seems natural to use the raethod used above to obtain estimates for pt. We have done this, and we obtain a relation of the form >/($,£), where the function on the right is different for n odd and n even. However, in both cases, lim/(s , í) = 2 (í — 1), which CO rendera the estimate useless (for, the /c-th prime is larger than 2(fc-1) for ali k). REFERENCES [1] O. GRUN, Ober ungerade volfkommene Zahlen, Math. Zeit, 55 (1952) pp. 353-354. [2] H.-J. KANOLD, Úber mehrfach volikommene Zahlen, J. f Ur Math. 194 (1955) pp. 218-220. [3] P. J. McCARTHY, Odd perfect numbers, to appear in Scripta Mathematica. [4] T. NAOEL, Introduction to Ntimber Theory, New York (1951). |
