Odd perfect numbers, Diophantine equations, and upper bounds. (English) Zbl 1325.11009
A classical problem in number theory asks if there are any odd perfect numbers. The author gives a new bound for the odd perfect numbers. Assume that \(N\) is an odd perfect number with \(k\) distinct prime divisors. Denote by \(P\) the largest prime divisor of \(N\). The author has formerly proved [Math. Comput. 76, No. 260, 2109–2126 (2007; Zbl 1142.11086)] that \(k\geq 9\) and \(N<2^{4^k}\). Combining theoretical results with computer calculations, in the present paper among others he shows, as a corollary of the main result, that \(10^{12}P^2N<2^{4^k}\).
Reviewer: István Gaál (Debrecen)
MSC:
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11Y50 | Computer solution of Diophantine equations |
| 11Y70 | Values of arithmetic functions; tables |
Citations:
Zbl 1142.11086References:
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