On finiteness of odd superperfect numbers

Yamada, Tomohiro

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 259-274.

On montre de nouveaux résultats sur l’équation $\sigma \left(N\right)=aM$, $\sigma \left(M\right)=bN.$ 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.

Some new results concerning the equation $\sigma \left(N\right)=aM$, $\sigma \left(M\right)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.

Classification : 11A25, 11A05, 11D61, 11J86
Mots clés : Odd perfect numbers, multiperfect numbers, superperfect number; the sum of divisors, arithmetic functions, exponential diophantine equations.

[1] Bugeaud, Yann; Győry, Kálmán Bounds for the solutions of Thue-Mahler equations and norm form equations, Acta Arith., Volume 74 (1996) no. 3, pp. 273-292 | DOI | MR | Zbl

[2] Coates, John An effective $p$-adic analogue of a theorem of Thue. II: The greatest prime factor of a binary form, Acta Arith., Volume 16 (1969), pp. 399-412 | DOI | MR | Zbl

[3] Dandapat, G. G.; Hunsucker, John L.; Pomerance, Carl Some new results on odd perfect numbers, Pac. J. Math., Volume 57 (1975), pp. 359-364 | DOI | MR | Zbl

[4] Dickson, Leonard E. Finiteness of the odd perfect and primitive abundant numbers with $n$ distinct prime factors, American J., Volume 35 (1913), pp. 413-422 | MR | Zbl

[5] Evertse, Jan-Hendrik On equations in $S$-units and the Thue-Mahler equation, Invent. Math., Volume 75 (1984), pp. 561-584 | DOI | MR | Zbl

[6] Evertse, Jan-Hendrik The number of solutions of the Thue–Mahler equation, J. Reine Angew. Math., Volume 482 (1997), pp. 121-149 | MR

[7] Fel’dman, Naum I. Improved estimate for a linear form of the logarithms of algebraic numbers, Mat. Sb., Volume 77 (1968), pp. 423-436 translation in Math. USSR, Sb. 6 (1968), p. 393-406

[8] Kotov, Sergey V. Greatest prime factor of a polynomial, Mat. Zametki, Volume 13 (1973), pp. 515-522 translation in Math. Notes 13 (1973), p. 313-317 | MR | Zbl

[9] Matveev, E. M. An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 64 (2000) no. 6, pp. 125-180 translation in Izv. Math. 64 (2000), no. 6, p. 127–169 | MR | Zbl

[10] Nielsen, Pace P. An upper bound for odd perfect numbers, Integers, Volume 3 (2003), A14, 9 pages | MR | Zbl

[11] Nielsen, Pace P. Odd perfect numbers, diophantine equation, and upper bounds, Math. Comput., Volume 84 (2015), pp. 2549-2567 | DOI | MR | Zbl

[12] Pomerance, Carl On multiply perfect numbers with a special property, Pac. J. Math., Volume 57 (1975), pp. 511-517 | DOI | MR | Zbl

[13] Pomerance, Carl Multiply perfect numbers, Mersenne primes and effective computability, Math. Ann., Volume 226 (1977), pp. 195-206 | DOI | MR | Zbl

[14] Schinzel, Andrzej On two theorems of Gelfond and some of their applications, Acta Arith., Volume 13 (1967), pp. 177-236 | DOI | MR | Zbl

[15] Schmidt, Wolfgang A. Diophantine approximations and diophantine equations, Lecture Notes in Mathematics, 1467, Springer, 1991 | MR | Zbl

[16] Shorey, Tarlok N.; Tijdeman, Robert Exponential diophantine equations, Cambridge Tracts in Mathematics, 87, Cambridge University Press, 1986 | MR | Zbl

[17] Suryanarayana, D. Super perfect numbers, Elem. Math., Volume 24 (1969), pp. 16-17 | MR | Zbl

[18] Suryanarayana, D. There is no odd super perfect number of the form $p^{2\alpha }$, Elem. Math., Volume 28 (1973), pp. 148-150 | MR | Zbl

[19] Yamada, Tomohiro Problem 005:10, 2005 (Western Number Theory Problems, http://www.ma.utexas.edu/users/goddardb/WCNT11/problems2005.pdf)

[20] Yamada, Tomohiro Unitary super perfect numbers, Math. Pannonica, Volume 19 (2008), pp. 37-47 | MR | Zbl

[21] Zsigmondy, Karl Zur Theorie der Potenzreste, Monatsh. Math., Volume 3 (1892), pp. 265-284 | DOI | MR | Zbl