Abstract. We show that there is an effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent.

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.

We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x p−1 − 1.where p is a prime, α 1 , α 2 are integers not divisible by p and b 1 , b 2 are integers with gcd(b 1 , b 2 , p) = 1. This result has been refined by several papers such as Schinzel [10], Yu [13][14][15][16][17], Bugeaud [2], and Bugeaud and Laurent [1]. Our purpose is to improve a result in the last paper by ✩ This paper is a revised version of my master thesis (Yamada, 2006 [12]).

We shall show that, for any positive integer D > 0 and any primes p 1 , p 2 not dividing D, the diophantine equation x 2 + D = 2 s p k 1 p l 2 has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}. IntroductionIt is known that the equation x 2 + 7 = 2 n has five solutions, as conjectured by Ramanujan and shown by Nagell [26] and other authors. According to this history, this diophantine equation has been called the Ramanujan-Nagell equation and several authors have studied various analogues.Apéry [1] showed that, for each integer D > 0 and prime p, the equation x 2 + D = p n has at most two solutions unless (p, D) = (2, 7) and, for any odd prime p, the equation x 2 +D = 4p n , which is equivalent to y 2 +y +(D +1)/4 = p n with y odd, also has at most two solutions. Beukers [5] showed that, if D > 0 and x 2 + D = 2 n has two solutions, then D = 23 or D = 2 k − 1 for some k > 3 and also gave an effective upper bound: if w = x 2 + D = 2 n with D = 0, then w < 2 435 |D| 10 . Further generalizations have been made by Le [18][19][20], Skinner [29] andBender and Herzberg [2] to prove that, for any given integers A, B, s, p with gcd(A, B) = 1, s ∈ {0, 2} and p prime, Ax 2 + B = 2 s p n has at most two solutions except 2x 2 + 1 = 3 k , 3x 2 + 5 = 2 k , x 2 + 11 = 4 × 3 k , x 2 + 19 = 4 × 5 k with three solutions and the Ramanujan-Nagell one x 2 + 7 = 2 k with five solutions.Bender and Herzberg [2] also found some necessary conditions for the equation D 1 x 2 + D 2 = 2 s a n with D 1 > 0, D 2 > 0, gcd(D 1 , D 2 ) = gcd(D 1 D 2 , k) = 1, s ∈ {0, 2} to have more than 2 ω(a) solutions. With the aid of the primitive divisor theorem of Bilu, Hanrot and Voutier [7] concerning Lucas and Lehmer sequences, Bugeaud and Shorey[10] determined all cases D 1 x 2 + D 2 = 2 m a n with D 1 > * 2010 Mathematics Subject Classification: 11D61 (Primary), 11D45 (Secondary).† Key words and phrases: Exponential diophantine equation, Ramanujan-Nagell equation, values of quadratic polynomials.

L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 32 (2020), 259-274 On finiteness of odd superperfect numbers par Tomohiro YAMADA Résumé. On montre de nouveaux résultats sur l'équation σ(N) = aM , σ(M) = bN. On en déduit, comme corollaire, qu'il n'existe qu'un nombre fini de nombres impairs superparfaits ayant un nombre fixé de facteurs premiers distincts.