NEW MULTIPLY PERFECT NUMBERS

ACHIM FLAMMENKAMP

Abstract. This paper summarizes the author’s results of a search for multiply perfect numbers. Especially some with multiplicity 9 have been discovered.

1. Introduction

For each natural number n, we write σ(n) for the sum of its divisors. If ∃k ∈ N : σ(n) = kn , then n is called a multiply perfect number and k its multiplicity. The case k = 1 is trivial, n = 1 is the only solution, and in the case k = 2 it is common known that n is a perfect number. The even perfect numbers are of the form n = 2^{p−1}(2^{p} − 1) , if 2^{p} − 1 is a prime, and it is conjectured that there are no odd ones [8]. Upto now 32 (bi-)perfect numbers has been computed, the largest with p = 756839 in 1992 by Gage and Slowinski.

2. On numbers with multiplicity > 2

2.1. History. For k > 2 there is a historical survey until 1919 on multiply perfect numbers in Dickson [1]. One of the latest there mentioned researchers, Cunningham[2], found in 1902 the first numbers with multiplicity 7. His work has been extended later by Franqui and García [3] and especially by Brown [4] who also found the first examples of muliplicity 8.

Besides, all found multiply perfect numbers, but 1, are even and it is an open question whether there are odd ones [5]. Further it is conjectured that in the case k = 2 there are infinite many multiply perfect numbers [9] in opposite for greater k there is supposed to be only finite many for each k; let us call this quantity a(k).

For k = 3 already Fermat in 1643 knew the 9 triperfect numbers which are the only ones known today. In the case k = 4 there seem to exist only 36 quadri-perfect which last Carmichael and Mason [6] published in 1910. For k > 4 I like to citated Madachy [7] who gave the numbers 62, 222, 217, 7 for the number of known 5-, 6and 7-perfect numbers in 1966. This gives lower bounds for the a(k). Upper bounds are not published upto day.

My aim was to find further examples of multiply perfect numbers, abbr. MPNs, especially for small k to get a probable complete list of these and to develop technics which can discover 9-perfect numbers.

1991 Mathematics Subject Classification. 11A25.

Key words and phrases. Multiply Perfect Numbers.

The author was supported by the Graduiertenkolleg Mathematik

2.2. Algorithmn. On effective computing methods there is an articel by Pommerance [10] but his algorithmn is not practically as he stated. Therefore I developed a recursive algorithmn to search for MPNs which can run on computers.

Look at each natural number n and at its prime factorization ∏_{i=1}^{l} p_{i}^{e_{i}}. For each prime power p_{i}^{e_{i}} σ(n) contains a factor (p_{i}^{e_{i}+1} − 1)/(p_{i} − 1). Therefore I used a dynamical updated table of factorizations of p_{i}, p_{i}^{2}, ..., p_{i}^{m} ^{1} with m as large as neccessary for given p_{i}. Then for multiplicity k and given number max you search for numbers n with fixed e_{0} in the range 1, ..., max(k). From the factorization of 2^{e_{0}+1} − 1 you get the first primes in σ(n) and therefore in kn, too. Having, say p_{j}^{e} , then e_{j} must have at least the value e if there exists a corresponding MPN. Now assume n containing a factor p_{j}^{e_{j}}, where e_{j} takes values upto a heuristic limit depending on p_{j}, e_{0} and k. So you get more and more primes p_{i} of n until you are forced to exceed a previously assumed e_{i} or you fits all the exponents and have calculated a k-perfect number.

2.3. Results. In the cases k = 3 or 4 I found no further MPNs but for k = 5 I computed 65 numbers. I believe that these three numbers are the values of those a(k). For k = 6 I extended the list of MPNs upto 245. The largest one is 2^{92}3^{18}5^{10}7^{4}13^{3}17^{4}19 · 29 · 43^{2}47^{3}79 · 127 · 167 · 199 · 281 · 467 · 607 · 631 · 701 · 1213 · 1523 · 1597 · 2141 · 2801 · 8563 · 9137 · 11093 · 36389 · 44371 · 88741 · 363889 · 12207031 · 29189717·2147483647·350432068432741·658812288653553079. For k = 7 I invested most of computing power, but I’m sure that a(7) > 484, the number of MPNs I produced. In the case k = 8 I discovered only 222 MPNs in spite of the existence of surely hundreds of further unknown MPNs. I give the remarkable smallest example of these: 2^{47}3^{26}5^{9}7^{10}11^{4}13^{5}17·19·23·29·31^{3}37^{2}43^{2}47·61·67^{2}71·73·79^{2}97·109·137· 157·179·241·257·281·337·379·433·521·631·673·757·821·1123·3221·8209·293459. Finally after about 50 cpu hours I found the first 9-perfect number of now three known: 2^{139}3^{43}5^{20}7^{9}11^{6}13^{7}17^{2}19^{6}23^{3}29·31^{3}37^{3}41^{2}43^{4}47·53·59·67·71·73·79·83·97· 103·107·109·113·127·131·137·167·191·193·223·257·281·307·331·379·431·467·643· 661·683·701·1039·1499·1723·2801·3067·3851·5501·6829·11807·14281·19531·45319· 70841·86171·122921·519499·570461·583367·3500201·7416361·23696191·47392381.

3. Acknowledgement

To realize this project, I used about 2000 hours of cputime on two DEC 5000/33, a SUN SPARC10, a IBM RS6000/320H, a HP 710 and a SGI Crimson. I am gratefully to the Fakultät für Mathematik and the Hochschul-Rechenzentrum of the University of Bielefeld which support my work with the power of this six computers.

References

1. L. E. Dickson, History of the Theory of Numbers, 1 (1913), 33–38.

2. A. Cunningham, , British Association Reports (1902), 35–36.

3. B. Franqui and M. García, 57 New Multiply Perfect Numbers, Scripta Math. 20 (1954), 169– 171

4. A. L. Brown, Multiperfect Numbers, Scripta Math. 20 (1954), 103–106.

5. M. M. Artuhov, On the problem on odd h-fold perfect numbers, Acta Arith. 23 (1973), 249– 255.

6. R. D. Carmichael and T. E. Mason, , Proc. Indiana Acad. Sc., (1911), 257–270.

1Thanks to A.K. Lenstra who placed his ECM-factorization implementation to my disposal.

7. J. S. Madachy, Madachy’s Mathematical Recreations, Dover Pub. Inc., New York, (1979), 150–151.

8. P. Ribenboim, The Little Book of Big Primes, Springer Verlag (1991), 65–75

9. S. S. Wagstaff, Divisors of Mersenne Numbers, Math. of Computation 40 (1983), 385–397.

10. C. Pommerance, Multiply Perfect numbers, Mersenne primes and effective computability, Math. Ann. 226 (1977), 195–206.

Fakultät für Mathematik, Universität Bielefeld, 33601 Bielefeld, Germany E-mail address: achim@hrz.uni-bielefeld.de