Download Odd Triperfect Numbers - American Mathematical Society

mathematics
volume
january
of computation
42, number
1984, pages
165
331-233
Odd Triperfect Numbers
By Masao Kishore*
Abstract. We prove that an odd triperfect number has at least ten distinct prime factors.
1. A positive number TVis called a triperfect number if a(N) — 37V,where o(/V) is
the sum of the positive divisors of TV.Six even triperfect numbers are known:
214• 5 • 7 ■19 • 31 - 151,
213- 3 • 11 • 43 • 127,
29-3 • 11 -31,
28 • 5 • 7 • 19 • 37 • 73,
25 • 3 • 7,
23 • 3 ■5.
However, the existence of an odd triperfect (OT) number is an open question.
McDaniel [1] and Cohen [2] proved that an OT number has at least nine distinct
prime factors, and Beck and Najar [3] showed that it exceeds IO50.
In this paper using the technique of [4], we prove
Theorem. // jV is OT, N has at least ten distinct prime factors.
2. Throughout this paper we let
N-YÍPÍf
i-i
where p, 's are odd primes, p, < • • • < pr and a, 's are positive integers.
The following lemmas are easy to prove:
Lemmal.IfNisOT,
(1)
a/s are even for 1 < I < r.
Lemma 2. // TVis OT and q is a prime factor of a(pf-) for some i, then q = 3 or
q = pj for some j, 1 ^ y < r.
Lemma3. If N is OT and r = 9, ps < 80.
ReceivedJuly 6, 1981.
1980 Mathematics Subject Classification. Primary 10A20.
* This paper was written as a quarter project when the author was in a programming language class
taught by Mr. Joseph Klinger at the University of Toledo in Spring, 1981.
©1984 American Mathematical Society
0025-5718/84 $1.00 + $.25 per page
231
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232
MASAO KISHORE
As in [4] we define
S(N) = a(N)/N,
a(p)
= Min(a|aisevenand/7a+1
> 10"},
b: = Min{a,, a(Pi)),
Lemma4. IfN is OT, then
(2)
log3 -r-
IO"" < logS(/tf)<log3.
Proof. Since M\ N, S(M) < S(N) = 3 and so log S(M) < log3. In [4] we proved
that if a > a(p), then
0 < \ogS(pa) - logSip«"))
< IO"".
Hence
0<logS(/V)-logS(M)<r-
10-",
and we have
log3 - r- 10"" =\ogS(N)-r-
IO'" < logS(M).
Q.E.D.
Corollary. IfN is OT, L = M/pb/ and ifpr > 3500,then
(3)
log3 - r ■10"" - IogS(34992) < logS(L) < log3.
Proof of Theorem. We used a computer (PDP 11 at the University of Toledo) to
find
M=Y\pf'
7=1
satisfying (1), a¡ < a( p¡) for 1 < ; < 9, (2) with r = 9, and pg < 3500. There were 71
such M's; however, all of them had a factor /?,"'such that a, < a(pt), o(pf') had a
prime factor <?> 3, and q * />-,1 < y < 9.
Next we tried to find
;=1
satisfying (1), a, < a(/0 for 1 «s / < 8, and (3) with r = 9. There were 12689 such
L's; however, 12473 of them had a factor p°' such that a¡ < a(p¡), a(p°')
had a
prime factor q > 3, q * pj for 1 < y < 8, and
logS(L) + logS(?2)>log3.
The remaining 216 of them had the following properties: there exist two consecutive primes u and v such that 3500 < u < v,
logS(L) + logS(u2) > log3,
and
logS(L) + \ogv/(v - 1) < log3 - 9 • IO"".
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ODD TRIPERFECT NUMBERS
233
These three cases show that if TVis an odd integer with r = 9, then TVcannot be
OT. Q.E.D.
The computer time was over five hours.
Department of Mathematics
East Carolina University
Greenville, North Carolina 27834
1. W. McDaniel, "On odd multiplyperfect numbers,"Boll. Un. Mat. hat.. No. 2,1970, pp. 185-190.
2. G. L. Cohen, "On odd perfect numbers II, Multiperfect numbers and quasiperfect numbers," J.
Austral.Math. Soc.,v. 29, 1980,pp. 369-384.
3. W. E. Beck & R. M. Najar, "A lower bound for odd triperfects," Math. Comp., v. 38, 1982,pp.
249-251.
4. M. Kishore. "Odd integers N with five distinct prime factors for which 2 - IO"12< a(N)/N < 2
+ IO''2," Math. Comp.,v. 32, 1978,pp. 303-309.
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