An exponential diophantine equation related to odd perfect numbers

Abstract

We shall show that, for any given primes ℓ ≥ 17 and p, q ≡ 1 (mod ℓ), the diophantine equation (x ℓ − 1)/(x − 1) = p m q has at most four positive integral solutions (x, m) and give its application to odd perfect number problem.

Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form

“…But they overlooked numbers of the form 3 α (q 1 q 2 • • • q r−1 ) 2 , which is our main concern of this paper. Now we would like to extend our results in [33], [35], and [36] for odd perfect numbers in the form (1.2) into odd k-perfect numbers by showing that there are only finitely many odd k-perfect numbers in the form (1.2) for any fixed k and β. But our method cannot be extended into odd k-perfect numbers in all cases.…”

“…Then, proceeding as in the proof of Lemma 3.2 of [35], we have δ + 2(#T − δ ≤ 2β(2β + t) + Ω 1 (k) or, in other words, #T ≤ β(2β + t) + (δ + Ω 1 (k))/2. From [36], we can deduce that δ ≤ 4#S ≤ 4(2β + t) and therefore #T ≤ (β + 2)(2β + t) + Ω 1 (k)/2. Thus, instead of (5.7) and (5.8), we have (5.9) #U < respectively.…”

“…Assume that an integer N in the form (1.1) is k-perfect. Since the case k = 2 has been settled by [36], we assume that k > 2. Moreover, assume that k = k 1 ℓ t for some integers t ≥ 0 and k 1 ≡ 0, 1 (mod ℓ) if 2β + 1 = ℓ g .…”

“…It is not difficult to show that if an integer N of the form (1.2) is k-perfect and, in the case 2β + 1 = ℓ g with ℓ a prime and g ≥ 1 an integer, k = k 1 ℓ t for integers t ≥ 0 and k 1 ≡ 0, 1 (mod ℓ), then we can take a prime divisor of 2β + 1 other than q r . Using methods employed by [33], [35], and [36], it follows that Theorem 1. However, this argument does not work if 2β + 1 = q g r and k = k 1 q t r for some integers g ≥ 1, t ≥ 0, and k 1 ≡ 1 (mod q r ).…”

“…Gathering more recent results such as [4], [7], [10], [16], and [17], we know that if N is an odd perfect number of the form (1.2), then β ≥ 9, β ≡ 1 (mod 3) , β ≡ 2 (mod 5), and β cannot take some other values such as 11, 14, 18, 24. The author proved that r ≤ 4β 2 + 2β + 3 and N < 2 4 4β 2 +2β+3 for an odd perfect number N in the form (1.2) in [33]. Later, the author replaced the upper bound for r by 2β 2 + 8β + 3 in [35] and then 2β 2 + 6β + 3 in [36].…”