Odd triperfect numbers

Abstract

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.

Odd triperfect numbers are divisible by twelve distinct prime factors

“…Hence 5 5 1 a(p^) or 5 5 6 . In (1), 3 2 + a(7 fl3 ), 3 + a(13 a '), 3 + a(19 a ')> 3 + a(31" 8 ) because 371 a(7 8 ), 611 <T(13 2 ), 1271 a(19 2 ), and 3311 a(31 2 ).…”

“…Since 829|a(5 8 ), we have 5 8 tt N by Lemma 1. Suppose that 5 6 \\N. Then p 10 = 19531 because a(5 6 ) = 19531.…”

“…LEMMA 21.176 + N.PROOF. By Corollary 9.2, 17 8 + N. Suppose that 176 ||JV. Since a(17 6 ) = 25646167, it follows that 256461671 iV.…”

“…McDaniel [8] and Cohen [4] proved that if N is OT, then u(N) > 9, where u(N) is the number of distinct prime factors of N. The author [6] proved that u(N) > 10 and [7] that a(N) > 11; Bugulov [3] also proved that «(JV) > 11. Beck and Najar [2] showed that N > 10 50 , and Alexander [1] proved that N > 10 60 .…”

Odd Triperfect Numbers are Divisible by Eleven Distinct Prime Factors

“…McDaniel [4] and Cohen [2] proved that an OT number has at least nine distinct prime factors; the author proved that it has at least ten prime factors [3], and Beck and Najar [1] showed that it exceeds 1050.…”