I just have a quick inquiry about the Wolfram MathWorld entry on Odd Perfect Number.

Last sentence of the fifth paragraph states that:

Hagis (1980) showed that odd perfect numbers must have at least eight distinct prime factors, in which case, the number is divisible by 15 (Voight 2003).

Since Nielsen (2015) has already proved that odd perfect numbers must have at least ten distinct prime factors, does this mean that they must be divisible by $15$?

Upon inspecting Voight's paper, I was under the impression that the divisibility-by-5 condition only holds for odd perfect numbers with exactly eight distinct prime factors.

Can anybody confirm if my impression is correct?

References

Hagis, P. Jr. "An Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors." Math. Comput. 34, 1027-1032, 1980.

Voight, J. "On the Nonexistence of Odd Perfect Numbers." MASS Selecta. Providence, RI: Amer. Math. Soc., pp. 293-300, 2003.

Nielsen, P. P. "Odd perfect numbers, Diophantine equations, and upper bounds", Math. Comp. 84, 2549-2567, 2015.

  • 1
    No, the above statement just means when we have EXACTLY $8$ prime factors, then, the number must be divisble by $15$. (This is at least how I would interprete it). Note : I did not read the whole question before posting this comment, I made this interpretation independent from the statement you mentioned about the exactly $8$ factors. – Peter Aug 27 at 17:23
  • @Peter: Kindly flesh out your last comment into an actual answer, so that this question does not go unanswered, and so that I would be able to accept it. Thank you very much! =) – Jose Arnaldo Bebita Dris Oct 18 at 14:31

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