Computo, ergo sum
I came across the following 12 digit number:
Before proceeding, the header image is from the following site:
https://www.famousscientists.org/rene-descartes/
Unfortunately I'm unable to recollect how I encountered this number, most likely in conjunction with "Perfect Numbers", though I have to admit that I have not felt interested in this topic, and precisely speaking "complete lack of interest in Perfect Numbers" would be a better expression in spite of being a kind of numberphile. So it's unlikely that I searched for perfect numbers, which is why I did not choose a commercially better sounding title "In search of an Odd Perfect Number" like my favorite book I read long time ago…
"À la recherche de nombres parfaits impairs inconnus" would be only allowed for a magnum opus, so it's not an option.
Of course, I was aware of the topic especially for the struggle of finding an Odd Perfect Number (OPN).
Perhaps its numerological #数秘 resonance was the reason for my distancing myself from Perfect Numbers.
The definition of Perfect Number #完全数 is just easy at elementary school level, as particularly compared with that of "irregular prime" (非正則 #素数 ) which I mentioned in my recent article relative to 37:
A number is called a perfect number if the sum of proper divisors of the number (called its aliquot sum) is equal to itself.
Using a divisor (sigma) function, a number n is a perfect number if and only if σ₁(n) = 2n.
https://en.wikipedia.org/wiki/Perfect_number
It is famous that the following 4 perfect numbers were known to Greeks (Euclid, Nicomachus) 2000+ years ago: 6, 28, 496 and 8128
And (even) Perfect Numbers are of the form
where p is a prime number and
is also a prime number.
Such a prime number is called a Mersenne Prime and it's most likely unnecessary to remind the audience that the name GIMPS (Great Internet Mersenne Prime Search) comes from Marin Mersenne.
https://www.mersenne.org/
According to the Wikipedia (English) mentioned above, the first known European mention of the fifth perfect number 33550336 (= 2^12 × 8191) is a manuscript written between 1456 and 1461 by an unknown mathematician. The manuscript (Clm 14908), which might have been written in Regensburg, is in the collection of the Bayerische Staatsbibliothek (Bavarian State Library) in Munich, and it is stated that 8191 (= 2^13 - 1) is prime, in other words, Mersenne prime M₁₃.
This suggests that 8191 had been recognized as prime prior to its discovery by Marin Mersenne in 1644, though the following document casts skepticism on the legitimacy of what is written in Wikipedia.
https://t5k.org/notes/by_year.html
P.S. Regensburg is a city I did wish to visit... I happened to stay at Landsberg am Lech located 160+ km southwest of Regensburg back in October 1987 (before the Fall of the Berlin Wall), and for some reason I visited East Berlin in April 1988 using Interflug, so my old passport has a stamp of DDR, which may evoke dreadful memories as described in the Century Trilogy by Ken Follett for many …
Back to the number 198585576189, it definitely shows up in the context of OPN (Odd Perfect Number).
Skipping some detail, here is what René Descartes wrote in his letter to Marin Mersenne on 15 November 1638:
Mais je pense pouvoir démontrer qu'il n'y en a point de pairs qui soient parfaits, excepté ceux d'Euclide; et qu'il n'y en a point aussi d'impairs, si ce n'est qu'ils soient composés d'un seul nombre premier, multiplié par un carré dont la racine soit composée de plusieurs autres nombres premiers. Mais je ne vois rien qui empêche qu'il ne s'en trouve quelques-uns de cette sorte: car, par exemple, si 22021 était nombre premier, en le multipliant par 9018009, qui est un carré dont la racine est composée des nombres premiers 3, 7, 11 et 13, on aurait 198.585.576.189, qui serait nombre parfait. Mais, quelque méthode dont on puisse user, il faut beaucoup de temps pour chercher ces nombres, et peut-être que le plus court a plus de 15 ou 20 notes.
Then after 361 years, another intriguing number －22017965903 was identified...I'm not sure if this can be called a breakthrough though...by John Voight according to the Quanta magazine:
The following explains the both spoof odd numbers (intentionally containing wrong calculations, just in case):
The above Quanta Magazine also states:
After employing 20 parallel processors for three years, the team found all possible spoof numbers with factorizations of six or fewer bases — 21 spoofs altogether, including the Descartes and Voight examples — along with two spoof factorizations with seven bases. Searching for spoofs with even more bases would have been impractical — and extremely time-consuming — from a computational standpoint. Nevertheless, the group amassed a sufficient sample to discover some previously unknown properties of spoofs.
and here is an excerpt from the paper "Odd, Spoof Perfect Factorizations" by BYU Computational Number Theory Group:
It's interesting that Wikipedia in French refers to the finding by John Voight while English version does not (as of 9th July 2024):
Whether OPN exist or not is still an open problem.
According to Wolfram Mathworld, if such a number exists, it should be bigger than 10^1500 (Pascal Ochem and Michaël Rao (2012)).
Mathematics of Computation
This number is humongous, and I would not blame if one declared non-existence of OPN.
One thing I wish to mention is that it is potentially undecidable though we don't know. Let's recall Hilbert's tenth problem (1900):
Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.
In 1970, Yuri Vladimirovich Matiyasevich, building on earlier work of M. Davis, H. Putnam, and J. Robinson, showed that no such algorithm exists.
("Undecidability in number theory" (Bjorn Poonen))
P.S. The 3rd perfect number 496 is very important in superstring theory, a difficult theory beyond my capacity... I only know 496 = 248 x 2 where 248 is the dimension of exceptional Lie group E₈. □