# $2^{11} - 1$ and the mystery of Huldaricus Regius

While researching on Mersenne numbers, I often stumble upon statements of this nature (it is not verbatim):

Huldaricus Regius in 1536 proved that $$2^{11}-1$$ is not prime, providing a factorisation and shattering the common misconception that all numbers of the form $$2^p-1$$ are primes for $$p$$ prime.

This is of course an important step in the history of Mersenne Primes -- and this made me wonder the following. You can find his name in different variations (Huldalrichus Regius, Hudalrichus Regius, Ulricus Regius, etc) and in multiple sources: this table (English), in this brief history of Mersenne primes (Italian) and in this one too (Italian). Although I can't read French, this book (French) could also come in handy.

He seems an interesting figure, but I could find little to no information about him. Was he real? Does anyone know more about him? I'm asking this of course for historical interest, but also because Huldaricus Regius is ominously similiar to "Huldaricus Regius [x]", which in Latin could possibly hint to a title or a group of people -- depending on the truncated [x].

• Well that is just a conjecture of mine, I meant that it could have been a noun/adj/other forgotten by history; sometimes it does happen. @DescheleSchilder Jul 30 '21 at 8:52

Huldalrichus Regius seems to be the same person as Ulrich Regius, otherwise known as Ulrich Rieger, according to the Consortium of European Research Libraries (CERL) entry for him.

His book, VTRIVSQVE ARITHMETICES Epitome ex uarijs authoribus concinnata per HVdalrichum Regium was published in 1536, with reissues in 1543 and 1550. You can see a digital version here. The Worldcat entry also gives details of editions and locations of copies. I'm guessing that the title means something like Summary of both kinds of arithmetic, compiled from various experts by Ulrich Regius

It was common in that era for scholars to use latinate forms of their birth names, and it is sometimes hard to us to match the forms of the names. According to https://www.mathgenealogy.org/ one of my doctoral ancestors is "Regiomontanus", who was the same person as Johannes Müller, from Königsberg. If you know a tiny bit of Latin and German you see that his scholarly name is a Latin calque of the name of his birthplace. Similarly, one of Euler's doctoral ancestors was Philip Melanchthon, whose name is Greek calque of his birth name, Philipp Schwartzerdt.

• Ha! I shoud call myself Marcus Trimontanes (Driebergen=three mountains, though it could be that it means castles). Jul 30 '21 at 5:40
• I see, thank you! That is very interesting, although I studied this age before I was not aware of the whole name ordeal. I shall edit the question for different spellings of the name (Huldarichus, Huldalrichus, etc) so it can be found quicker. Jul 30 '21 at 9:01

Hudalrichus Regius (English spelling?) is perhaps best known as the discoverer of the first perfect number since Euclid, who had identified 6, 28, 496 and 8128.

Euclid knew that $$2^{n-1}(2^n - 1)$$ is perfect when $$(2^n - 1)$$ is prime and applied this to $$n = 2, 3, 5, 7$$. Hudalrichus Regius tried $$n = 11$$ and found that $$2^{11}-1$$ was not prime, however $$n = 13$$ does yield a prime and the corresponding perfect number $$33350336$$.

This also refuted a claim of Nicomachus that the $$n'\text{th}$$ perfect number had $$n$$ digits.

Source : MacTutor entry on Pietro Antonio Cataldi.

Source : MacTutor entry on perfect numbers.