In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, a Fermi–Dirac prime is a
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, ...
whose exponent is a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negati ...
. These numbers are named from an analogy to
Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
based on the fact that each integer has a unique representation as a product of Fermi–Dirac primes without repetition. Each element of the sequence of Fermi–Dirac primes is the smallest number that does not divide the product of all previous elements.
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
used the Fermi–Dirac primes to find the smallest number whose
number of divisors is a given power of two.
Definition
The Fermi–Dirac primes are a sequence of numbers obtained by raising a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
to an exponent that is a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negati ...
. That is, these are the numbers of the form
where
is a prime number and
is a non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. These numbers form the sequence:
They can be obtained from the prime numbers by repeated
squaring, and form the smallest set of numbers that includes all of the prime numbers and is
closed under squaring.
Another way of defining this sequence is that each element is the smallest positive integer that does not divide the product of all of the previous elements of the sequence.
Factorization
Analogously to the way that every positive integer has a unique
factorization
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...
, its representation as a product of prime numbers (with some of these numbers repeated), every positive integer also has a unique factorization as a product of Fermi–Dirac primes, with no repetitions allowed. For example,
The Fermi–Dirac primes are named from an analogy to
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and ...
. In physics,
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s are particles that obey
Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic eq ...
, in which it is allowed for multiple particles to be in the same state at the same time.
Fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s are particles that obey
Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
, which only allow a single particle in each state. Similarly, for the usual prime numbers, multiple copies of the same prime number can appear in the same prime factorization, but factorizations into a product of Fermi–Dirac primes only allow each Fermi–Dirac prime to appear once within the product.
Other properties
The Fermi–Dirac primes can be used to find the smallest number that has exactly
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s, in the case that
is a power of two,
. In this case, as
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
proved, the smallest number with
divisors is the product of the
smallest Fermi–Dirac primes. Its divisors are the numbers obtained by multiplying together any subset of these
Fermi–Dirac primes. For instance, the smallest number with 1024 divisors is obtained by multiplying together the first ten Fermi–Dirac primes:
In the theory of infinitary divisors of Cohen, the Fermi–Dirac primes are exactly the numbers whose only infinitary divisors are 1 and the number itself.
References
{{reflist, refs=
[{{citation, first=Daniel J., last=Bernstein, author-link=Daniel J. Bernstein, contribution=Enumerating and counting smooth integers, contribution-url=https://cr.yp.to/papers/epsi-19950518-retypeset20220326.pdf, title= Detecting Perfect Powers in Essentially Linear Time, and Other Studies in Computational Number Theory, type=Doctoral dissertation, publisher=University of California, Berkeley, year=1995]
[{{citation
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, pages = 395–411
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, volume = 54
, year = 1990, jstor = 2008701
; see especially Corollary 1, p. 401.]
[{{citation
, last = Grost , first = M. E.
, doi = 10.1080/00029890.1968.11971056
, journal = The American Mathematical Monthly
, jstor = 2315183
, mr = 234901
, pages = 725–729
, title = The smallest number with a given number of divisors
, volume = 75
, year = 1968, issue = 7
]
[{{citation
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, last2 = Shevelev , first2 = Vladimir
, journal = Integers
, mr = 2342191
, page = A33, 35
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, url = https://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html
, volume = 7
, year = 2007]
[See the closely related sequence {{cite OEIS, A084400, mode=cs2, which differs only in that it includes 1 at the start of the sequence. However, 1 does divide the ]empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in questio ...
of all previous elements.
[{{cite OEIS, A005179, Smallest number with exactly n divisors, mode=cs2]
[{{cite OEIS, A050376, Fermi-Dirac primes: numbers of the form p^(2^k) where p is prime and k ≥ 0, mode=cs2]
[{{cite OEIS, A037992, Smallest number with 2^n divisors, mode=cs2]
[{{citation
, last = Ramanujan , first = S. , author-link = Srinivasa Ramanujan
, doi = 10.1112/plms/s2_14.1.347
, issue = 1
, journal = Proceedings of the London Mathematical Society
, pages = 347–409
, title = Highly Composite Numbers
, volume = s2-14
, year = 1915; see section 47, pp. 405–406, reproduced in ''Collected Papers of Srinivasa Ramanujan'', Cambridge Univ. Press, 2015]
pp. 124–125
/ref>
[{{citation
, last = Shevelev , first = V. S.
, issue = 4
, journal = Izvestiya Vysshikh Uchebnykh Zavedeniĭ, Severo-Kavkazskiĭ Region, Estestvennye Nauki
, mr = 1647060
, pages = 28–43, 101–102
, title = Multiplicative functions in the Fermi–Dirac arithmetic
, year = 1996]
Prime numbers
Integer sequences