In number theory, a FermiāDirac prime is a prime power whose exponent is a power of two. These numbers are named from an analogy to FermiāDirac statistics in physics based on the fact that each integer has a unique representation as a product of FermiāDirac primes without repetition. Each element of the sequence of FermiāDirac primes is the smallest number that does not divide the product of all previous elements. Srinivasa Ramanujan used the FermiāDirac primes to find the smallest number whose number of divisors is a given power of two.
The FermiāDirac primes are a sequence of numbers obtained by raising a prime number to an exponent that is a power of two. That is, these are the numbers of the form where is a prime number and is a non-negative integer. These numbers form the sequence: [1]
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. [1]
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. [2]
Analogously to the way that every positive integer has a unique factorization, 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 FermiāDirac primes, with no repetitions allowed. [3] [4] For example,
The FermiāDirac primes are named from an analogy to particle physics. In physics, bosons are particles that obey BoseāEinstein statistics, in which it is allowed for multiple particles to be in the same state at the same time. Fermions are particles that obey FermiāDirac statistics, 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 FermiāDirac primes only allow each FermiāDirac prime to appear once within the product. [1] [5]
The FermiāDirac primes can be used to find the smallest number that has exactly divisors, [6] in the case that is a power of two, . In this case, as Srinivasa Ramanujan proved, [1] [7] the smallest number with divisors is the product of the smallest FermiāDirac primes. Its divisors are the numbers obtained by multiplying together any subset of these FermiāDirac primes. [7] [8] [9] For instance, the smallest number with 1024 divisors is obtained by multiplying together the first ten FermiāDirac primes: [8]
In the theory of infinitary divisors of Cohen, [10] the FermiāDirac primes are exactly the numbers whose only infinitary divisors are 1 and the number itself. [1]