Are All Perfect Numbers Even?/Progress/Prime Factors/Historical Note

Historical Note on Prime Factors of Odd Perfect Number

James Joseph Sylvester stated that there exist no odd perfect number with fewer than $6$ distinct prime factors, and proved that there are none with fewer than $8$ if none of those prime factors is $3$.

Bryant Tuckerman published a proof in $1973$ that an odd perfect number $P$ has the properties that:

either:

at least one of the prime powers factoring $P$ is greater than $10^{18}$

the power of such a prime factor is even

or:

there is no divisor of $P$ less than $7$.

Peter Hagis, Jr. published a proof in $1980$ that an odd perfect number has at least $8$ distinct prime factors.

Peter Hagis, Jr. and Graeme L. Cohen published a proof in $1998$ that an odd perfect number has at least one prime factor which is greater than $1 \, 000 \, 000$.

Pascal Ochem and Michaël Rao published a proof in $1998$ that:

at least one of the prime powers factoring an odd perfect number is greater than $10^{62}$

an odd perfect number has more than $101$ prime factor (not neessarily distinct).

Sources

• 1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$ ... (previous) ... (next): Preface

• 1973: Bryant Tuckerman: A Search Procedure and Lower Bound for Odd Perfect Numbers (Math. Comp. Vol. 27: pp. 943 – 949)  www.jstor.org/stable/2005529

• 1980: Peter Hagis, Jr.: An Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors (Math. Comp. Vol. 34: pp. 1027 – 1032)  www.jstor.org/stable/2006211

• 1998: Peter Hagis, Jr. and Graeme L. Cohen: Every Odd Perfect Number Has a Prime Factor Which Exceeds $10^6$ (Math. Comp. Vol. 67: pp. 1323 – 1330)  www.jstor.org/stable/2585187

• 2012: Pascal Ochem and Michaël Rao: Odd Perfect Numbers Are Greater than $10^{1500}$ (Math. Comp. Vol. 81: pp. 1869 – 1877)  www.jstor.org/stable/23268069