On the Second and Third Largest Prime Divisors of an Odd Perfect Number

Question

Over twenty years years ago, Douglas Iannucci proved that the second largest prime divisor of an OPN exceeds $10,000$.

See https://www.ams.org/journals/mcom/1999-68-228/S0025-5718-99-01126-6/S0025-5718-99-01126-6.pdf

(Incidentally, I believe that he also proved that the third largest prime divisor of an OPN exceeds $100$.)

Are these still the best known bounds to date or have either bound been increased?

Thank you.

Answer 1

You are correct, both bounds (proved by Douglas E. Iannucci) have not been improved.

These facts are noted in two (recent) separate papers:

Upper bounds on the second largest prime factor of an odd perfect number - by Joshua Zelinsky (2018) - page 2 - Iannucci (1999) proved that the second largest prime divisor of an odd perfect number exceeds $10000$.

On the third largest prime divisor of an odd perfect number - by Sean Bibby, Pieter Vyncke, and Joshua Zelinsky (2019) - page 2 - Iannucci (2000) proved that the third largest prime divisor of an odd perfect number exceeds $100$.

At least one of the authors (Zelinsky) has an MSE account.