Perfect numbers

First 10 perfect numbers

As the reader will have already realised, the history of perfect numbers is littered with errors and Cataldi , despite having made the major advance of finding two new perfect numbers, also made some false claims. In fact assuming that perfect numbers are of the form 2p-1 2p - 1 where p is prime, the question readily translates into asking whether - 1 is prime. In Pervusin showed that 1 is a perfect number. J Scheybl gave the sixth perfect number in in his commentary to a translation of Euclid 's Elements. However, since the time of Nicomachus we do know a lot more about his five assertions than the simplistic statement we have just made. Later in the same year he improved his result to five factors and, over the years, this has been steadily improved until today we know that an odd perfect number would have to have at least eight distinct prime factors, and at least 29 prime factors which are not necessarily distinct. I had to cheat to make the last perfect number display in regular form. Then in Euler settled the last of Cataldi 's claims when he proved that - 1 was not prime so Cataldi 's guesses had not been very good. He told Roberval in that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic. He accepted Nicomachus 's classification of numbers but the work is purely mathematical, not containing the moral comments of Nicomachus.

He accepted Nicomachus 's classification of numbers but the work is purely mathematical, not containing the moral comments of Nicomachus. Ibn al-Haytham proved a partial converse to Euclid 's proposition in the unpublished work Treatise on analysis and synthesis when he showed that perfect numbers satisfying certain conditions had to be of the form 2k-1 2k - 1 where 2k - 1 is prime.

Nicomachus goes on to describe certain results concerning perfect numbers.

largest perfect number

To join the search, look for information here. As we have seen this algorithm is precisely that given by Euclid in the Elements. It is perhaps surprising that Regius, who must have thought he had made one of the major breakthroughs in mathematics, is virtually unheard of today.

odd perfect numbers

In Kraitchik showed that Mersenne was wrong in his claims for his largest prime of when he showed that 1 is not prime. Without speaking a work he multiplied the two numbers together to get and sat down to applause from the audience.

It also leads to an easy proof that all even perfect numbers end in either a 6 or 8 but not alternately. The more computers join the search, the faster these numbers can be found.

mersenne primes and perfect numbers
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Perfect Number