Are All Triperfect Numbers Even?/Progress
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Progress on Open Question: Are All Triperfect Numbers Even?
While it is not known whether there exist any odd triperfect numbers, several important facts have been established.
Minimum Size of Odd Triperfect Number
It has been established that an odd triperfect number, if one were to exist, would be greater than $10^{70}$.
If it does not have $3$ as a prime factor, then it is greater than $10^{108}$.
Form of Odd Triperfect Number
An odd triperfect number is square.
Prime Factors of Odd Triperfect Number
An odd triperfect number has:
- at least $11$ distinct prime factors
- at least $32$ distinct prime factors if $3$ is not one of them.
Sources
- Jan. 1982: Walter E. Beck and Rudolph M. Najar: A Lower Bound for Odd Triperfects (Math. Comp. Vol. 38, no. 157: pp. 249 – 251) www.jstor.org/stable/2007481
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $120$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $120$