A new upper bound for odd perfect numbers of a special form

Abstract

We show that there is no odd perfect number of the form 2 n + 1 or n n + 1. * 2010 Mathematics Subject Classification: 11A05, 11A25. † Key words and phrases: Odd perfect numbers, sum of divisors, arithmetic functions.

An exponential diophantine equation related to odd perfect numbers

“…Combining this result with an argument in [21], we obtain the following new upper bound for odd perfect numbers of a special form.…”

“…We have shown that, if N = p α (q 1 q 2 • • • q k ) 2β is an odd perfect number, then k ≤ 4β 2 + 2β + 2 in [19]. Recently, we have improved this upper bound by 2β 2 + 8β + 3 in [21], where the coefficient 8 of β can be replaced by 7 if 2β + 1 is not a prime or β ≥ 29. Since it is known that N < 2 4 k+1 from [17], we have…”

“…Our method is similar to the approach used in [21]. In this paper, we use upper bounds for sizes of solutions of (1) derived from a Baker-type estimate for linear forms of logarithms by Matveev [13], which may be interesting itself, while [21] used an older upper bound for odd perfect numbers of the form given above.…”

“…Our method is similar to the approach used in [21]. In this paper, we use upper bounds for sizes of solutions of (1) derived from a Baker-type estimate for linear forms of logarithms by Matveev [13], which may be interesting itself, while [21] used an older upper bound for odd perfect numbers of the form given above. We note that Padé approximations using hypergeometric functions given by Beukers [2][3] does not work in our situation since our situation will give much weaker approximation to √ D, although Beukers' gap argument is still useful (see Lemma 2.4 below).…”

“…In the next section, we introduce some preliminary results from [21] and Matveev's lower bounds for linear forms of logarithms. In Section 3, using Matveev's lower bounds, upper bounds for the sizes of solutions of (1) is given.…”