Transcript Bounding the Factors of Odd Perfect Numbers

Bounding the Factors
of Odd Perfect Numbers
Charles Greathouse
Miami University
Perfect Numbers
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σ(n) is the sum of divisors function: σ(n) = d|n
Σd
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A number N is called perfect iff σ(N)=2N:
σ(6) = 1 + 2 + 3 + 6 = 2 · 6
 σ(496) = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
+ 496 = 2 · 496
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The 42nd known perfect number was discovered
in February of 2005. All 42 are even.
Odd Perfect Numbers
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Odd Perfect Conjecture: There exist no odd
perfect numbers.
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Euler: All perfect numbers are of the form pax²,
with the p≡a≡1 (mod 4) and gcd(p, x) = 1.
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This means that for an odd perfect n =
p1a1p2a2…ptat (with the pi distinct primes) all but
the “special prime” will have even exponents.
Abundancy
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σ-1(n) := d|n
Σ d-1 = σ(n) / n. σ-1(n) is clearly multiplicative
since σ(n) is multiplicative.
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Note that 1 < (p + 1) / p ≤ σ-1(pa) < p / (p – 1) and
σ-1(pa) < σ-1(pa+1). We can bound σ-1(pa) without knowing a.
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Since N is perfect iff σ-1(n) = 2, if we have
N=p1p2…pk…pm and σ-1(p1p2…pk) > 2 then n is not
perfect since σ-1 is strictly increasing on prime powers.
Bounding with Abundancy
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Suppose the odd perfect N = 3a5b7cx with
integral variables and {3, 5, 7, x} pairwise
relatively prime.
3 and 7 can’t be the special prime, so a,c ≥ 2.
σ-1(N) = σ-1(3a5b7cx) ≥ σ-1(325172x) = σ-1(32) σ-1(5)
σ-1(72) σ-1(x) = 13/9 · 6/5 · 57/49 · σ-1(x) ≥ 2.016.
No such N can ever be perfect!
Bounding with Abundancy
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So if 3, 5, and 7 cannot all divide odd perfect n,
we know that the third-smallest divisor of an
odd perfect number is at least 11.
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I will write N = p1a1p2a2…ptat with p1 < p2 < … <
pt for all odd perfect N, with the pi distinct
primes. Thus the above result is p3 ≥ 11.
Bounds for t = 9
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31
29
23
19
17
13
11
5
3
≤
≤
≤
≤
≤
≤
≤
≤
≤
p9
p8
p7
p6
p5
p4
p3
p2
p1
Literature
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Using previous notation:
 pt > 107 (Jenkins 2003)
 pt-1 > 104 (Iannucci 2000)
 pt-2 > 102 (Iannucci 1999)
 p1 = 3 when t < 11 (Hagis 1983 and Kishore
1983)
Bounds for t = 9
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107 ≤
104 ≤
101 ≤
19 ≤
17 ≤
13 ≤
11 ≤
5 ≤
3 ≤
p9
p8
p7
p6
p5
p4
p3
p2
p1
Literature
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Using previous notation:
 pt > 107 (Jenkins 2003)
 pt-1 > 104 (Iannucci 2000)
 pt-2 > 102 (Iannucci 1999)
 p1 = 3 when t < 11 (Hagis 1983 and Kishore
1983)
i-1
2
 pi < 2 (t-i+1) for 2 ≤ i ≤ 6 (Kishore 1981)
t
4
 N < 2 (Nielsen 2003)
Bounds for t = 9
 107
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9
4
2
< p9 < ≈ 1078914
9
4
4
10 < p8 < 2 ≈ 1078914
9
4
101 < p7 < 2 ≈ 1078914
5
2
19 ≤ p6 < 2 · 4 = 17 179 869 184
4
2
17 ≤ p5 < 2 · 5 = 327 680
3
2
13 ≤ p4 < 2 · 6 = 1536
2
2
11 ≤ p3 < 2 · 7 = 112
1
2
5 ≤ p2 < 2 · 8 = 32
p1 = 3
Bounds for t = 9
 107
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9
4
2
< p9 < ≈ 1078914
9
4
4
10 < p8 < 2 ≈ 1078914
9
4
101 < p7 < 2 ≈ 1078914
19 ≤ p6 ≤ 17 179 869 143
17 ≤ p5 ≤ 327 673
13 ≤ p4 ≤ 1531
11 ≤ p3 ≤ 109
5 ≤ p2 ≤ 31
p1 = 3
More with Abundncy
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So p2 ≤ 31, but can we improve this?
Suppose p2 = 31. Then σ-1(n) ≤ σ-1
(3a31b37c41d43e47f101g10007h10000019i) < 1.7254
 Even if p2 = 13 we have σ-1(n) ≤ σ-1
(3a13b17c19d23e29f101g10007h10000019i) < 1.9934
 We can use this technique to bound p3 as well, but
no further. Without other restrictions we have
1.905 < σ-1 (3a5b11c) < 2.084.

Bounds for t = 9
 107
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9
4
2
< p9 < ≈ 1078914
9
4
4
10 < p8 < 2 ≈ 1078914
9
4
101 < p7 < 2 ≈ 1078914
19 ≤ p6 ≤ 17 179 869 143
17 ≤ p5 ≤ 327 673
13 ≤ p4 ≤ 1531
11 ≤ p3 ≤ 31
5 ≤ p2 ≤ 11
p1 = 3
Additional Results
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There are a number of other, less general results
that can be used to narrow the bounds on odd
perfect numbers:
With care, we can tighten the bounds from Nielsen’s
result: it is a bound on the whole number and not its
individual factors.
 One known, these can be used as inputs for further
abundancy restrictions.
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Bounds for t = 9
 107
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9
4
2
< p9 < ≈ 1078914
9
4
4
10 < p8 < 2 ≈ 1026305
9
4
101 < p7 < 2 ≈ 1015783
29 ≤ p6 ≤ 17 179 869 143
19 ≤ p5 ≤ 327 673
13 ≤ p4 ≤ 1531
11 ≤ p3 ≤ 31
5 ≤ p2 ≤ 11
p1 = 3
t>9
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Most of the previous results can be extended in
some fashion to OPNs with more than 9
distinct primes.
Unfortunately, few methods can be generalized
sufficiently with current methods. These
computational results seem useful only for
extending various bounds, not for proving the
Odd Perfect Conjecture.
Skepticism
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Sylvester's Web of Conditions
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“…a prolonged meditation on the subject has
satisfied me that the existence of any one such—its
escape, so to say, from the complex web of
conditions which hem it in on all sides—would be
little short of a miracle.”
Skepticism
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Pomerance Heuristic
If n=pm² is odd perfect then p | σ(m²).
 Thus there are at most log m possibilities for p.
 The ‘probability’ that σ(n) is divisible by n is p/n =
1/m².
 The sum over m converges, so there ‘should’ exist at
most a finite number of odd perfects.
 Since there are no OPNs up to 10300, “it may be
more appropriate to sum (log m)/m² for m > 1075.”
 This is 10-70, so it is reasonable to conjecture that no
odd perfect numbers exist.
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Where from here?
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Kevin Hare has published a number of papers
recently restricting the number of total prime
factors (to a minimum of 75 in his latest
preprint).
Suryanarayana and Hagis have a paper discussing
the sum of reciprocals of the distinct prime
factors of OPNs.
If the methods of Kishore’s 1981 paper could
be extended to include more factors, stricter
bounds could be place on these numbers.
Bibliography
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Peter Hagis, Jr., “Sketch of a proof that an odd perfect number relatively prime to 3 has at least
eleven prime factors,” Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 399-404.
Douglas E. Iannucci, “The third largest prime divisor of an odd perfect number exceeds one
hundred,” Mathematics of Computation, Vol. 69, No. 230 (2000), pp. 867-879.
Douglas E. Iannucci, “The second largest prime divisor of an odd perfect number exceeds ten
thousand,” Mathematics of Computation, Vol. 68, No. 228 (1999), pp. 1749-1760.
Paul M. Jenkins, “Odd perfect numbers have a prime factor exceeding 107,” Mathematics of
Computation, Vol. 72, No. 243 (2003), pp. 1549-1554.
Masao Kishore, “Odd perfect numbers not divisible by 3. II,” Mathematics of Computation, Vol. 40,
No. 161 (Jan., 1983), pp. 405-411.
Masao Kishore, “On odd perfect, quasiperfect, and odd almost perfect numbers,” Mathematics of
Computation, Vol. 36, No. 154 (Apr., 1981), pp. 583-586.
William Lipp, “Odd Perfect Number Search,” http://www.oddperfect.org/.
Pace P. Nielsen, “An upper bound for odd perfect numbers,” INTEGERS: Electronic Journal of
Combinatorial Number Theory, Vol. 3 (2003), #A14.