Bounding the Factors of Odd Perfect Numbers Charles Greathouse Miami University.

Bounding the Bounding the FactorsFactors

of Odd Perfect of Odd Perfect NumbersNumbers

Charles GreathouseCharles Greathouse

Miami UniversityMiami University

Perfect NumbersPerfect Numbers

σσ(n) is the sum of divisors function (n) is the sum of divisors function σσ(n) (n) = = ΣΣdd

A number N is called A number N is called perfectperfect iff iff σσ(N)=2N(N)=2N σσ(6) = 1 + 2 + 3 + 6 = 2 (6) = 1 + 2 + 3 + 6 = 2 6 6 σσ(496) = 1 + 2 + 4 + 8 + 16 + 31 + 62 + (496) = 1 + 2 + 4 + 8 + 16 + 31 + 62 +

124 + 248 + 496 = 2 124 + 248 + 496 = 2 496 496

The 42The 42ndnd known perfect number was known perfect number was discovered in February of 2005 All 42 discovered in February of 2005 All 42 are evenare even

d|n

Odd Perfect NumbersOdd Perfect Numbers

Odd Perfect Conjecture There exist no Odd Perfect Conjecture There exist no odd perfect numbersodd perfect numbers

Euler All perfect numbers are of the Euler All perfect numbers are of the form pform paaxsup2 with the pequivaequiv1 (mod 4) and xsup2 with the pequivaequiv1 (mod 4) and gcd(p x) = 1gcd(p x) = 1

This means that for an odd perfect n = This means that for an odd perfect n = pp11

aa11pp22aa22hellipphellipptt

aatt (with the p (with the pii distinct primes) distinct primes) all but the ldquospecial primerdquo will have all but the ldquospecial primerdquo will have even exponentseven exponents

AbundancyAbundancy

σσ-1-1(n) = (n) = ΣΣ dd-1-1 = = σσ(n) n (n) n σσ-1-1(n) is clearly (n) is clearly multiplicative since multiplicative since σσ(n) is multiplicative(n) is multiplicative

Note that 1 lt (p + 1) p le Note that 1 lt (p + 1) p le σσ-1-1(p(paa) lt p (p ndash 1) ) lt p (p ndash 1) and and σσ-1-1(p(paa) lt ) lt σσ-1-1(p(pa+1a+1) ) We can bound We can bound σσ-1-1(p(paa)) without knowing without knowing aa

Since N is perfect iff Since N is perfect iff σσ-1-1(n) = 2 if we have (n) = 2 if we have N=pN=p11pp22hellipphellippkkhellipphellippmm and and σσ-1-1(p(p11pp22hellipphellippkk) gt 2 then n is ) gt 2 then n is not perfect since not perfect since σσ-1-1 is strictly increasing on is strictly increasing on prime powersprime powers

d|n

Bounding with Bounding with AbundancyAbundancy

Suppose the odd perfect N = 3Suppose the odd perfect N = 3aa55bb77ccx with x with integral variables and 3 5 7 x pairwise integral variables and 3 5 7 x pairwise relatively primerelatively prime

3 and 7 canrsquot be the special prime so ac ge 3 and 7 canrsquot be the special prime so ac ge 22

σσ-1-1(N) = (N) = σσ-1-1(3(3aa55bb77ccx) ge x) ge σσ-1-1(3(32255117722x) = x) = σσ-1-1(3(322) ) σσ--

11(5) (5) σσ-1-1(7(722) ) σσ-1-1(x) = 139 (x) = 139 65 65 5749 5749 σσ-1-1(x) ge (x) ge 20162016

No such N can ever be perfectNo such N can ever be perfect


So if 3 5 and 7 cannot all divide odd So if 3 5 and 7 cannot all divide odd perfect n we know that the third-perfect n we know that the third-smallest divisor of an odd perfect smallest divisor of an odd perfect number is at least 11number is at least 11

I will write N = pI will write N = p11aa11pp22

aa22hellipphellippttaatt with p with p11 lt lt

pp2 2 lt hellip lt plt hellip lt ptt for all odd perfect N with for all odd perfect N with the pthe pii distinct primes Thus the above distinct primes Thus the above result is presult is p33 ge 11 ge 11

Bounds for t = 9Bounds for t = 9 31 le p31 le p99

29 le p29 le p88

23 le p23 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11

LiteratureLiterature

Using previous notationUsing previous notation pptt gt 10 gt 1077 (Jenkins 2003) (Jenkins 2003) ppt-1t-1 gt 10 gt 1044 (Iannucci 2000) (Iannucci 2000) ppt-2t-2 gt 10 gt 1022 (Iannucci 1999) (Iannucci 1999) pp11 = 3 when t lt 11 (Hagis 1983 and = 3 when t lt 11 (Hagis 1983 and

Kishore 1983)Kishore 1983)

Bounds for t = 9Bounds for t = 9 101077 le p le p99

101044 le p le p88

101 le p101 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11



Kishore 1983)Kishore 1983) ppii lt 2 lt 222i-1i-1

(t-i+1) for 2 le i le 6 (Kishore (t-i+1) for 2 le i le 6 (Kishore 1981)1981)

N lt 2N lt 244tt (Nielsen 2003) (Nielsen 2003)

Bounds for t = 9Bounds for t = 9 101077 lt p lt p99 lt 2 lt 24499 asymp 10 asymp 107891478914

101044 lt p lt p88 lt 2 lt 24499 asymp 10 asymp 107891478914

101101 lt plt p77 lt 2 lt 24499 asymp 10 asymp 107891478914

19 le p19 le p66 lt 2 lt 22255 4 = 17 179 869 184 4 = 17 179 869 184 17 le p17 le p55 lt 2 lt 22244 5 = 327 680 5 = 327 680 13 le p13 le p44 lt 2 lt 22233 6 = 1536 6 = 1536 11 le p11 le p33 lt 2 lt 22222 7 = 112 7 = 112 5 le p5 le p22 lt 2 lt 22211 8 = 32 8 = 32 pp1 1 = 3 = 3




19 le p19 le p66 le 17 179 869 143 le 17 179 869 143 17 le p17 le p55 le 327 673 le 327 673 13 le p13 le p44 le 1531 le 1531 11 le p11 le p33 le 109 le 109 5 le p5 le p22 le 31 le 31 pp1 1 = 3 = 3

More with AbundncyMore with Abundncy

So pSo p22 le 31 but can we improve this le 31 but can we improve this Suppose pSuppose p22 = 31 Then = 31 Then σσ-1-1(n) le (n) le σσ-1 -1

(3(3aa3131bb3737cc4141dd4343ee4747ff101101gg1000710007hh1000001910000019ii) lt ) lt 1725417254

Even if pEven if p22 = 13 we have = 13 we have σσ-1-1(n) le (n) le σσ-1 -1


We can use this technique to bound pWe can use this technique to bound p33 as as well but no further Without other well but no further Without other restrictions we haverestrictions we have1905 lt 1905 lt σσ-1 -1 (3(3aa55bb1111cc) lt 2084) lt 2084





Additional ResultsAdditional Results

There are a number of other less There are a number of other less general results that can be used to general results that can be used to narrow the bounds on odd perfect narrow the bounds on odd perfect numbersnumbers With care we can tighten the bounds With care we can tighten the bounds

from Nielsenrsquos result it is a bound on from Nielsenrsquos result it is a bound on the whole number and not its individual the whole number and not its individual factorsfactors

One known these can be used as inputs One known these can be used as inputs for further abundancy restrictionsfor further abundancy restrictions




29 le p29 le p66 le 17 179 869 143 le 17 179 869 143 19 le p19 le p55 le 327 673 le 327 673 13 le p13 le p44 le 1531 le 1531 11 le p11 le p33 le 31 le 31 5 le p5 le p22 le 11 le 11 pp11 = 3 = 3

t gt 9t gt 9

Most of the previous results can be Most of the previous results can be extended in some fashion to OPNs extended in some fashion to OPNs with more than 9 distinct primeswith more than 9 distinct primes

Unfortunately few methods can be Unfortunately few methods can be generalized sufficiently with current generalized sufficiently with current methods These computational methods These computational results seem useful only for results seem useful only for extending various bounds not for extending various bounds not for proving the Odd Perfect Conjectureproving the Odd Perfect Conjecture

SkepticismSkepticism

Sylvesters Web of ConditionsSylvesters Web of Conditions ldquohellipldquohellipa prolonged meditation on the a prolonged meditation on the

subject has satisfied me that the subject has satisfied me that the existence of any one suchmdashits escape existence of any one suchmdashits escape so to say from the complex web of so to say from the complex web of conditions which hem it in on all sidesmdashconditions which hem it in on all sidesmdashwould be little short of a miraclerdquowould be little short of a miraclerdquo

SkepticismSkepticism Pomerance HeuristicPomerance Heuristic

If n=pmsup2 is odd perfect then p | If n=pmsup2 is odd perfect then p | σσ(msup2)(msup2) Thus there are at most log m possibilities for Thus there are at most log m possibilities for

pp The lsquoprobabilityrsquo that The lsquoprobabilityrsquo that σσ(n) is divisible by n is (n) is divisible by n is

pn = 1msup2pn = 1msup2 The sum over m converges so there lsquoshouldrsquo The sum over m converges so there lsquoshouldrsquo

exist at most a finite number of odd perfectsexist at most a finite number of odd perfects Since there are no OPNs up to 10Since there are no OPNs up to 10300300 ldquoit may ldquoit may

be more appropriate to sum (log m)msup2 for m be more appropriate to sum (log m)msup2 for m gt 10gt 107575rdquordquo

This is 10This is 10-70-70 so it is reasonable to conjecture so it is reasonable to conjecture that no odd perfect numbers existthat no odd perfect numbers exist

Where from hereWhere from here

Kevin Hare has published a number of Kevin Hare has published a number of papers recently restricting the number papers recently restricting the number of total prime factors (to a minimum of of total prime factors (to a minimum of 75 in his latest preprint)75 in his latest preprint)

Suryanarayana and Hagis have a paper Suryanarayana and Hagis have a paper discussing the sum of reciprocals of discussing the sum of reciprocals of the distinct prime factors of OPNsthe distinct prime factors of OPNs

If the methods of Kishorersquos 1981 paper If the methods of Kishorersquos 1981 paper could be extended to include more could be extended to include more factors stricter bounds could be place factors stricter bounds could be place on these numberson these numbers

BibliographyBibliography Peter Hagis Jr ldquoSketch of a proof that an odd perfect number relatively Peter Hagis Jr ldquoSketch of a proof that an odd perfect number relatively

prime to 3 has at least eleven prime factorsrdquo prime to 3 has at least eleven prime factorsrdquo Mathematics of ComputationMathematics of Computation Vol 40 No 161 (Jan 1983) pp 399-404Vol 40 No 161 (Jan 1983) pp 399-404

Douglas E Iannucci ldquoThe third largest prime divisor of an odd perfect number Douglas E Iannucci ldquoThe third largest prime divisor of an odd perfect number exceeds one hundredrdquo exceeds one hundredrdquo Mathematics of ComputationMathematics of Computation Vol 69 No 230 (2000) Vol 69 No 230 (2000) pp 867-879pp 867-879

Douglas E Iannucci ldquoThe second largest prime divisor of an odd perfect Douglas E Iannucci ldquoThe second largest prime divisor of an odd perfect number exceeds ten thousandrdquo number exceeds ten thousandrdquo Mathematics of ComputationMathematics of Computation Vol 68 No 228 Vol 68 No 228 (1999) pp 1749-1760(1999) pp 1749-1760

Paul M Jenkins ldquoOdd perfect numbers have a prime factor exceeding 10Paul M Jenkins ldquoOdd perfect numbers have a prime factor exceeding 1077rdquo rdquo Mathematics of ComputationMathematics of Computation Vol 72 No 243 (2003) pp 1549-1554 Vol 72 No 243 (2003) pp 1549-1554

Masao Kishore ldquoOdd perfect numbers not divisible by 3 IIrdquo Masao Kishore ldquoOdd perfect numbers not divisible by 3 IIrdquo Mathematics of Mathematics of ComputationComputation Vol 40 No 161 (Jan 1983) pp 405-411 Vol 40 No 161 (Jan 1983) pp 405-411

Masao Kishore ldquoOn odd perfect quasiperfect and odd almost perfect Masao Kishore ldquoOn odd perfect quasiperfect and odd almost perfect numbersrdquo numbersrdquo Mathematics of ComputationMathematics of Computation Vol 36 No 154 (Apr 1981) pp Vol 36 No 154 (Apr 1981) pp 583-586583-586

William Lipp ldquoOdd Perfect Number Searchrdquo httpwwwoddperfectorgWilliam Lipp ldquoOdd Perfect Number Searchrdquo httpwwwoddperfectorg Pace P Nielsen ldquoAn upper bound for odd perfect numbersrdquo Pace P Nielsen ldquoAn upper bound for odd perfect numbersrdquo INTEGERS INTEGERS

Electronic Journal of Combinatorial Number TheoryElectronic Journal of Combinatorial Number Theory Vol 3 (2003) A14 Vol 3 (2003) A14

Bounding the Factors of Odd Perfect Numbers

Perfect NumbersPerfect Numbers

σσ(n) is the sum of divisors function (n) is the sum of divisors function σσ(n) (n) = = ΣΣdd

A number N is called A number N is called perfectperfect iff iff σσ(N)=2N(N)=2N σσ(6) = 1 + 2 + 3 + 6 = 2 (6) = 1 + 2 + 3 + 6 = 2 6 6 σσ(496) = 1 + 2 + 4 + 8 + 16 + 31 + 62 + (496) = 1 + 2 + 4 + 8 + 16 + 31 + 62 +

124 + 248 + 496 = 2 124 + 248 + 496 = 2 496 496

The 42The 42ndnd known perfect number was known perfect number was discovered in February of 2005 All 42 discovered in February of 2005 All 42 are evenare even

d|n







AbundancyAbundancy




d|n





11(5) (5) σσ-1-1(7(722) ) σσ-1-1(x) = 139 (x) = 139 65 65 5749 5749 σσ-1-1(x) ge (x) ge 20162016








29 le p29 le p88

23 le p23 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11





101044 le p le p88

101 le p101 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11
































t gt 9t gt 9

































AbundancyAbundancy




d|n





11(5) (5) σσ-1-1(7(722) ) σσ-1-1(x) = 139 (x) = 139 65 65 5749 5749 σσ-1-1(x) ge (x) ge 20162016








29 le p29 le p88

23 le p23 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11





101044 le p le p88

101 le p101 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11
































t gt 9t gt 9































11(5) (5) σσ-1-1(7(722) ) σσ-1-1(x) = 139 (x) = 139 65 65 5749 5749 σσ-1-1(x) ge (x) ge 20162016








29 le p29 le p88

23 le p23 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11





101044 le p le p88

101 le p101 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11
































t gt 9t gt 9

































29 le p29 le p88

23 le p23 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11





101044 le p le p88

101 le p101 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11
































t gt 9t gt 9































101044 le p le p88

101 le p101 le p77

19 le p19 le p66

17 le p17 le p55

13 le p13 le p44

11 le p11 le p33

5 le p5 le p22

3 le p3 le p11
































t gt 9t gt 9


























































t gt 9t gt 9



























Bounding the Factors of Odd Perfect Numbers Charles Greathouse Miami University.

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Transcript of Bounding the Factors of Odd Perfect Numbers Charles Greathouse Miami University.