TIL about the “Fermi-Dirac primes”, which mostly aren’t primes, and which AFAIK neither Fermi nor Dirac took any interest in. Ramanujan was interested in them, but he certainly didn’t name them this way, as Dirac was only 18 when Ramanujan died.
The Fermi-Dirac primes are integers that are primes raised to a power that is in turn a power of 2, i.e. p^{2^k}.
In order, they are:
2, 3, 4, 5, 7, 9, 11, 13, 16, ...
Every positive integer can be factored uniquely into a product of powers of primes, with no restriction on the powers of each prime.
But every positive integer can also be factored uniquely into a product of Fermi-Dirac primes, with each FDP appearing *at most once* in the product. Hence the name: the factors here are analogous to particles obeying Fermi-Dirac statistics, where each quantum state is occupied by either 0 or 1 particles, no more.
For example:
56 = 2^3 × 7 = 2^1 × 2^2 × 7 = 2 × 4 × 7
Ramanujan proved that the smallest number with exactly 2^k divisors is the product of the first k FDPs. For example:
2 × 3 × 4 × 5 × 7 × 9 = 7560
is the smallest number with exactly 2^6 = 64 divisors (including 1 and itself), and you can enumerate them simply by forming all 2^6 products that either include or omit each of those six FDPs.
Interesting. The fact that every number can be factored in FD primes is easy to see, since the normal primes are a subset of the FD primes (as p^{2^0}). And since any natural number can be expressed as the sum of powers of 2 (this is the base^* of the binary number system), we can decompose whatever power of p is needed into the product of different powers of 2. Nice!
Huh. This is so *almost* but not quite related to some of the stuff I was poking around recently--constructible regular polygons, particularly. (The odd-sided ones have numbers of sides whose prime factorizations are square-free, with unique primes of the form 2^(2^n) + 1. But those really have to be primes.)
The code for generating these in OEIS looks slow and bad, but if you have a streaming generator for the usual primes, then the Fermi–Dirac primes are easy to generate in sorted order: just merge the primes and the squares of the recursively generated Fermi–Dirac primes.
🗓️ Dimanche 18 mai à 18 h au port du Rosmeur (#Douarnenez).
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Le collectif #Palestine Douarnenez crie ÇA SUFFIT !
Das Bild zeigt eine Instagram-Post von "comey" mit einem Foto von Sand, auf dem die Zahlen "8647" aus verschiedenen Muscheln und Steinen gebildet sind. Die Zahlen sind in verschiedenen Farben und Größen von Muscheln und Steinen angeordnet, die auf dem Sand liegen. Der Hintergrund besteht aus weichem, grauem Sand mit einigen Fußabdrücken und kleinen Steinen. Die Muscheln und Steine sind in verschiedenen Farben, darunter Braun, Grau und Schwarz. Der Text des Posts lautet: "comey Cool shell formation on my beach walk 1 hour ago" und zeigt 71 Likes.
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Trump says “nothing happens” on Ukraine until he meets Putin. While Europe pushes for sanctions after failed talks, Trump stalls, leaving allies guessing — again. No pressure on the Kremlin.
mstdn.social/@noelreports/1145… It seems that the only option is to send a very clear invite to Putin. Drop the bridge. Europe needs to help by providing some of the means to do it. Putin may realize that his game is over when that happens.
Trump says “nothing happens” on Ukraine until he meets Putin. While Europe pushes for sanctions after failed talks, Trump stalls, leaving allies guessing — again. No pressure on the Kremlin.
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i know, trump is taking too much of my attention already and i'm not even us-american. but once in a while i need the reminder that trump is not only horrible but also a clown: a real live horror clown. and then i feel somewhat lucky i don't live in the usa and feel very much afraid because the usa are very contagious.
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By Palestine Chronicle Staff More than 100 Palestinians were killed, injured, or remain missing following heavy Israeli airstrikes on northern Gaza’s residential areas and medical services. At dawn on Friday, Israeli forces launched a [...]
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Stephan Schulz
in reply to Greg Egan • • •Interesting. The fact that every number can be factored in FD primes is easy to see, since the normal primes are a subset of the FD primes (as p^{2^0}). And since any natural number can be expressed as the sum of powers of 2 (this is the base^* of the binary number system), we can decompose whatever power of p is needed into the product of different powers of 2. Nice!
*Is this a pun? 😉
Matt McIrvin
in reply to Greg Egan • • •This is so *almost* but not quite related to some of the stuff I was poking around recently--constructible regular polygons, particularly. (The odd-sided ones have numbers of sides whose prime factorizations are square-free, with unique primes of the form 2^(2^n) + 1. But those really have to be primes.)
0xDE
in reply to Greg Egan • • •