Timeline for Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 28, 2014 at 7:49 | vote | accept | Jose Arnaldo Bebita | ||
| Aug 13, 2014 at 15:13 | comment | added | Jose Arnaldo Bebita | Fletcher, Nielsen and Ochem have a relatively recent paper where they show, using a new factor chain argument, that $5$ does not divide an odd perfect number indivisible by a sixth power. | |
| Aug 13, 2014 at 15:05 | comment | added | Jose Arnaldo Bebita | Thank you very much for your detailed comments @Jan-ChristophSchlage-Puchta. Appreciate it! | |
| Aug 13, 2014 at 13:53 | comment | added | Jan-Christoph Schlage-Puchta | For example, McDaniel (On the divisibility of an odd perfect number by the sixth power of a prime. Math. Comp. 25 (1971), 383–385) showed that an odd perfect number is either divisible by the sixth power of a prime, or it is not divisible by any prime $<100$. The proof is quite easy, once one has shown that if $n$ is not divisible by the sixth power of a prime, then $n$ is not divisible by some small prime, but getting there is pretty difficult. | |
| Aug 13, 2014 at 13:49 | comment | added | Jan-Christoph Schlage-Puchta | There is a lot of literature proving the non-existence of odd perfect numbers satisfying certain additional restrictions. Practically all of them require some starting point of the form "One of these primes divides $n$" or "One of these primes does not divide $n$". Sometimes this starting point is easy to get (e.g. if you bound the number of distinct prime factors), but quite often getting the starting point is the most difficult part. | |
| Aug 13, 2014 at 13:29 | comment | added | Jan-Christoph Schlage-Puchta | Cohen and Hagis proved that the largest prime factor of an odd perfect number is $>10^6$. Of course, the proof is indirect, so it begins with "Let $n$ be an odd perfect number with all prime divisors $\leq 10^6$...". Then a sequence of properties for $n$ are deduced, one of which reads "Lemma: $n$ is not divisible by 3, 5, 7, ..." . This Lemma was later taken out of context and cited as "Let $n$ be an odd perfect number. Then $n$ is not divisible by 3,5,7,...". | |
| Aug 12, 2014 at 13:08 | comment | added | Jose Arnaldo Bebita | Additionally, can you comment more on why you say that "such a result would be a major breakthrough regarding our knowledge on odd perfect numbers"? | |
| Aug 12, 2014 at 13:06 | comment | added | Jose Arnaldo Bebita | Thank you for your answer @Jan-ChristophSchlage-Pu. I guess what Cohen and Hagis really proved is that the largest prime factor of an odd perfect number exceeds ${10}^6$? | |
| Aug 12, 2014 at 11:54 | history | answered | Jan-Christoph Schlage-Puchta | CC BY-SA 3.0 |