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number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Mills' constant is defined as the smallest positive
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
''A'' such that the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
of the double exponential function : \left\lfloor A^ \right\rfloor is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
for all positive
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s ''n''. This constant is named after William Harold Mills who proved in 1947 the existence of ''A'' based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unproven, but if the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
is true, it is approximately 1.3063778838630806904686144926... .


Mills primes

The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins :2, 11, 1361, 2521008887, 16022236204009818131831320183, :4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, \ldots . If ''ai'' denotes the ''i'' th prime in this sequence, then ''ai'' can be calculated as the smallest prime number larger than a_^3. In order to ensure that rounding A^, for ''n'' = 1, 2, 3, ..., produces this sequence of primes, it must be the case that a_i < (a_+1)^3. The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large cube numbers, which is sufficient to prove this inequality if we start from a sufficiently large first prime a_1. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the ''sufficiently large'' condition to be removed, and allowing the sequence of Mills primes to begin at ''a''1 = 2. For all a > e^, there is at least one prime between a^3 and (a+1)^3. This upper bound is much too large to be practical, as it is infeasible to check every number below that figure. However, the value of Mills' constant can be verified by calculating the first prime in the sequence that is greater than that figure. As of April 2017, the 11th number in the sequence is the largest one that has been ''proved'' prime. It is :\displaystyle (((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220 and has 20562 digits. , the largest known Mills ''probable'' prime (under the Riemann hypothesis) is :\displaystyle (((((((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220)^3+66768)^3+300840)^3+1623568)^3+8436308 , which is 1,665,461 digits long.


Numerical calculation

By calculating the sequence of Mills primes, one can approximate Mills' constant as :A\approx a(n)^. Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills' constant under the assumption that the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
is true. Mills' constant is not known to have a closed-form formula, but it is known to be
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
.


Generalisations

There is nothing special about the middle exponent value of 3. It is possible to produce similar prime-generating functions for different middle exponent values. In fact, for any real number above 2.106..., it is possible to find a different constant ''A'' that will work with this middle exponent to always produce primes. Moreover, if Legendre's conjecture is true, the middle exponent can be replaced with value 2 . Matomäki showed unconditionally (without assuming Legendre's conjecture) the existence of a (possibly large) constant ''A'' such that \lfloor A^ \rfloor is prime for all ''n''. Additionally, Tóth proved that the floor function in the formula could be replaced with the
ceiling function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
, so that there exists a constant B such that :\lceil B^ \rceil is also prime-representing for r>2.106\ldots. In the case r=3, the value of the constant B begins with 1.24055470525201424067... The first few primes generated are: :2, 7, 337, 38272739, 56062005704198360319209, 176199995814327287356671209104585864397055039072110696028654438846269, \ldots ''Without'' assuming the Riemann hypothesis, Elsholtz proved that \lfloor A^ \rfloor is prime for all positive integers , where A \approx 1.00536773279814724017, and that \lfloor B^ \rfloor is prime for all positive integers , where B \approx 3.8249998073439146171615551375.


See also

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Formula for primes In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what ...


References


Further reading

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External links

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Who remembers the Mills number?
E. Kowalski.
Awesome Prime Number Constant
Numberphile. {{Prime number classes Mathematical constants Prime numbers