Hagis and McDaniel have shown that the largest prime factor of an odd perfect number TV is at least 100111, and Pomerance has shown that the second largest prime factor is at least 139. Using these facts together with the method we develop, we show that if 3 | N, N is divisible by at least ten distinct primes.
Abstract. We prove that an odd triperfect number has at least ten distinct prime factors. The following lemmas are easy to prove:Lemma l.IfNisOT,(1) a/s are even for 1 < I < r.Lemma 2. // TV is OT and q is a prime factor of a(pf-) for some i, then q = 3 or q = pj for some j, 1 ^ y < r.Lemma 3. If N is OT and r = 9, ps < 80.
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