On finiteness of odd superperfect numbers
Tomohiro Yamada1
1 Center for Japanese language and culture, Osaka University, 562-8558, 8-1-1, Aomatanihigashi, Minoo, Osaka, Japan
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 259-274.

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

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1121
Classification: 11A25,  11A05,  11D61,  11J86
Keywords: Odd perfect numbers, multiperfect numbers, superperfect number; the sum of divisors, arithmetic functions, exponential diophantine equations.
Tomohiro Yamada 1

1 Center for Japanese language and culture, Osaka University, 562-8558, 8-1-1, Aomatanihigashi, Minoo, Osaka, Japan 
@article{JTNB_2020__32_1_259_0,
     author = {Tomohiro Yamada},
     title = {On finiteness of odd superperfect numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {259--274},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1121},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1121/}
}
TY  - JOUR
TI  - On finiteness of odd superperfect numbers
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 259
EP  - 274
VL  - 32
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1121/
UR  - https://doi.org/10.5802/jtnb.1121
DO  - 10.5802/jtnb.1121
LA  - en
ID  - JTNB_2020__32_1_259_0
ER  - 
%0 Journal Article
%T On finiteness of odd superperfect numbers
%J Journal de Théorie des Nombres de Bordeaux
%D 2020
%P 259-274
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1121
%R 10.5802/jtnb.1121
%G en
%F JTNB_2020__32_1_259_0
Tomohiro Yamada. On finiteness of odd superperfect numbers. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 259-274. doi : 10.5802/jtnb.1121. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1121/

[1] Yann Bugeaud; Kálmán Győry Bounds for the solutions of Thue-Mahler equations and norm form equations, Acta Arith., Volume 74 (1996) no. 3, pp. 273-292 | Article | MR: 1373714 | Zbl: 0861.11024

[2] John Coates 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 | Article | MR: 263741 | Zbl: 0221.10026

[3] G. G. Dandapat; John L. Hunsucker; Carl Pomerance Some new results on odd perfect numbers, Pac. J. Math., Volume 57 (1975), pp. 359-364 | Article | MR: 384681 | Zbl: 0295.10005

[4] Leonard E. Dickson Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American J., Volume 35 (1913), pp. 413-422 | MR: 1506194 | Zbl: 44.0220.02

[5] Jan-Hendrik Evertse On equations in S-units and the Thue-Mahler equation, Invent. Math., Volume 75 (1984), pp. 561-584 | Article | MR: 735341 | Zbl: 0521.10015

[6] Jan-Hendrik Evertse The number of solutions of the Thue–Mahler equation, J. Reine Angew. Math., Volume 482 (1997), pp. 121-149 | MR: 1427659

[7] Naum I. Fel’dman 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] Sergey V. Kotov Greatest prime factor of a polynomial, Mat. Zametki, Volume 13 (1973), pp. 515-522 (translation in Math. Notes 13 (1973), p. 313-317) | MR: 327670 | Zbl: 0267.12002

[9] E. M. Matveev 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: 1817252 | Zbl: 1013.11043

[10] Pace P. Nielsen An upper bound for odd perfect numbers, Integers, Volume 3 (2003), A14, 9 pages | MR: 2036480 | Zbl: 1085.11003

[11] Pace P. Nielsen Odd perfect numbers, diophantine equation, and upper bounds, Math. Comput., Volume 84 (2015), pp. 2549-2567 | Article | MR: 3356038 | Zbl: 1325.11009

[12] Carl Pomerance On multiply perfect numbers with a special property, Pac. J. Math., Volume 57 (1975), pp. 511-517 | Article | MR: 379349 | Zbl: 0304.10004

[13] Carl Pomerance Multiply perfect numbers, Mersenne primes and effective computability, Math. Ann., Volume 226 (1977), pp. 195-206 | Article | MR: 439730 | Zbl: 0324.10001

[14] Andrzej Schinzel On two theorems of Gelfond and some of their applications, Acta Arith., Volume 13 (1967), pp. 177-236 | Article | MR: 222034 | Zbl: 0159.07101

[15] Wolfgang A. Schmidt Diophantine approximations and diophantine equations, Lecture Notes in Mathematics, Volume 1467, Springer, 1991 | MR: 1176315 | Zbl: 0754.11020

[16] Tarlok N. Shorey; Robert Tijdeman Exponential diophantine equations, Cambridge Tracts in Mathematics, Volume 87, Cambridge University Press, 1986 | MR: 891406 | Zbl: 0606.10011

[17] D. Suryanarayana Super perfect numbers, Elem. Math., Volume 24 (1969), p. 16-17 | MR: 237424 | Zbl: 0165.36001

[18] D. Suryanarayana There is no odd super perfect number of the form p 2α , Elem. Math., Volume 28 (1973), pp. 148-150 | MR: 330035 | Zbl: 0266.10011

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

[20] Tomohiro Yamada Unitary super perfect numbers, Math. Pannonica, Volume 19 (2008), pp. 37-47 | MR: 2426782 | Zbl: 1164.11004

[21] Karl Zsigmondy Zur Theorie der Potenzreste, Monatsh. Math., Volume 3 (1892), pp. 265-284 | Article | MR: 1546236 | Zbl: 24.0176.02

Cited by Sources: