Wikipedia:Reference desk/Archives/Mathematics/2020 July 25

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July 25

Fifth perfect number

We know that the sequence of perfect numbers starts 6, 28, 496, 8128, 33550336. But some sources say 2096128 rather than 33550336 is the fifth perfect number. Any reason they do this?? Georgia guy (talk) 15:38, 25 July 2020 (UTC)

Per the Euclid–Euler theorem, All even perfect numbers can be written as $2^{p-1}*(2^{p}-1)$, but any number of this kind is only perfect if $2^{p}-1$ is prime. While $2096128=2^{10}*(2^{11}-1)$ those sources you're looking at mistakenly believe that 2047 is prime, when it is equal to $23*89$. Iffy★Chat -- 15:52, 25 July 2020 (UTC)

Do lots of sources make similar mistakes on lots of numbers?? Georgia guy (talk) 16:17, 25 July 2020 (UTC)

Sources get maths wrong regularly, but the great thing about maths is that you can discount mathematical errors from even the most reliable sources, all you need is the proof. Iffy★Chat -- 18:13, 25 July 2020 (UTC)

What often happens is that if an error occurs in a popular source then it will be repeated by others who are using that source as a reference, and this happens more often when the people doing the repeating aren't trained mathematicians themselves. In this case the original error is apparently due to Eric Temple Bell in his book The Last Problem, see [1]. --RDBury (talk) 19:18, 25 July 2020 (UTC)

A bibliographic reference for the short article linked to documenting Bell's error is: Edward T. Frankel. "Bell's imperfect perfect numbers". Fibonacci Quarterly 15(4):336, December 1977. It is interesting that Bell lists 130816 as the fifth perfect number, so if he originated the 2096128 error in the chain of copied errors, some copier noticed the 130816 error but did not thoroughly check the rest, even though they now also should have been considered iffy. Benjamin Franklin listed the perfect numbers in Poor Richard Improved for the year 1749 as being "6, 28, 120, &c.",^[2] so it appears he was not aware of the fact that a number of the form $2^{p{-}1}(2^{p}-1)$ is abundant if $p$ is composite. The same two imperfect perfect numbers as Bell's can be found, not by copying but as an apparently repeated original sin, in a 1974 letter to the editor: Eleanor H. McKeeman. The Arithmetic Teacher 21(4):308,328,335.^[3] The errors are noticed, though, in a reply in the same issue immediately following the letter. McKeeman's letter shows that she too was not aware of the need for $p$ to be prime. So I think Bell may have been neither the first nor the last imperfect number theorist in chains of copied errors leading to claims that 2096128 is the fifth perfect number.  --Lambiam 07:35, 26 July 2020 (UTC)

I didn't notice Bell had an erroneous entry in addition to 2096128, making it less likely that Bell was the source of the errors that the OP was talking about. The idea was to apply Occam's razor to explain why two sources my have the same error; two people making the same error independently seems more unlikely than one of them using the other as a source. It depends on the error though since not recognizing that 2^p-1 must be prime (as I assume you meant to say above) seems the kind of error that people are likely to make independently. It's interesting that Franklin had yet a different error in his list, at least he cited is sources though, and he did correct his error in a later edition according to one of the footnotes in the text. His almanac was published only two years after Euler's result was made public, and his source was published before that. Given the evidence it seems unlikely that Franklin knew about Euler's result, or if he did then he wasn't applying it correctly. (Fun fact, Franklin and Euler were born less than 2 years apart.) As an educated gentleman of the 18th century, Franklin would, of course, have studied Euclid; I don't know if the number theory part was on the standard curriculum though. --RDBury (talk) 10:10, 26 July 2020 (UTC)

Even if Franklin had read Elements IX.36 as a teenager, it is unlikely he remembered the details when he was 43. Euler's result is about all even perfect numbers while Euclid's is only about numbers of the form $2^{p{-}1}(2^{p}-1)$, but I cannot imagine Euclid not realizing that his sufficiency condition, when restricted to this class of numbers, is also necessary. This should in fact be obvious to anyone who understood Euclid's proof.  --Lambiam 14:29, 26 July 2020 (UTC)

Hmm, the lack of machinery available to Euclid (e.g. no algebraic notation) makes his sufficiency proof awkward and hard to follow; in my copy it runs nearly three pages. So to me it's hard so say what Euclid may or may not have realized. We shall never know for sure since all that was written down was the sufficiency proof. In any case, it's seems most likely that Franklin was relying on Edmund Stone rather than Euclid directly. Heath, in my copy of the elements, references Dickson's History of the Theory of Numbers, which has a more detailed list of who knew what when, and according to it, (see the Preface) by the time that Stone was writing it was well established that the fifth perfect number is 33550336, even without Euler's converse. Dickson says: "Very many early writers believed that 2^p-1 is a prime for every odd value of p." But this was refuted in 1536. Perhaps Stone was getting his information either directly or indirectly from one of these early writers and this may be relevant to the original question as well. --RDBury (talk) 22:45, 26 July 2020 (UTC)