Mills' constant

Mills' constant is defined in number theory as the smallest positive real number , so rounding off the double exponential function ${\displaystyle A}$

${\displaystyle \lfloor A^{3^{n}}\rfloor }$

results in a prime number for all positive integers (this means the rounding function ). The constant was named after William H. Mills who proved its existence in 1947 and relied on the work of Guido Hoheisel and Albert Ingham on prime number gaps. The exact value of the constant is unknown, but if the Riemann hypothesis is true, it is about 1.3063778838630806904686144926… (sequence A051021 in OEIS ). ${\displaystyle n}$ ${\displaystyle \lfloor \cdot \rfloor }$

Mills prime numbers

The primes generated by Mills 'constant are known as Mills' prime numbers. If the Riemann hypothesis is true, this sequence begins with:

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ... ( continuation A051254 in OEIS ).

If the i -th denotes the prime number of the sequence, then the smallest prime number can be calculated as greater . To ensure that rounding for n = 1, 2, 3, ... produces a sequence of prime numbers, must also apply. The Hoheisel-Ingham estimate guarantees that there is always a prime between any two sufficiently large cube numbers, which is sufficient to prove this inequality for a sufficiently large first prime number . Since the Riemann hypothesis implies that there is a prime number between any two consecutive cube numbers, the restriction of “sufficiently large” numbers can be dropped, resulting in the smallest Mills prime number of a ₁ = 2. ${\displaystyle a_{i}}$ ${\displaystyle a_{i}}$ ${\displaystyle a_{i-1}^{3}}$ ${\displaystyle A^{3^{n}}}$ ${\displaystyle a_{i}<(a_{i-1}+1)^{3}}$ ${\displaystyle a_{1}}$

The 11th and largest currently known Mills prime is:

${\displaystyle (((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220}$

It has 20,562 positions and was discovered by François Morain on June 5, 2006 . However, it was only proven in April 2017 that this number is actually a prime number.

At the moment 3 other Mills prime numbers are known (assuming the Riemann hypothesis). Should the hypothesis not be correct, these three numbers are at least PRP numbers .

The 14th and largest currently known Mills prime (assuming the Riemann hypothesis) is:

${\displaystyle \displaystyle ((((((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220)^{3}+66768)^{3}+300840)^{3}+1623568}$

It has 555,154 positions.

The number of digits roughly triples for each additional Mills prime number.

The following sequence of numbers (for ) generates these prime numbers using : ${\displaystyle b(n)}$ ${\displaystyle n=1,2,3,...}$ ${\displaystyle a(n)}$ ${\displaystyle b(n)=a(n+1)-a(n)^{3}}$

3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568 (follow A108739 in OEIS )

Numerical calculation

Mills' constant can be approximated by calculating the Mills prime numbers as follows:

${\displaystyle A\approx a(n)^{1/3^{n}}}$

With this method, Caldwell and Cheng were able to calculate the constant with an accuracy of 6850 decimal places. It is not known whether Mills' constant can be calculated in a closed form, nor is it a rational number . If it is rational and if you know the period of the decimal representation of this rational number, you can generate an infinite number of prime numbers with it (see prime number generator ).

Fractional approximation of Mills' constant

Mills' constant can also be represented approximately by continued fractions . The continued fraction representation of are: ${\displaystyle A}$ ${\displaystyle A\approx 1{,}3063778838630806904686144926\ldots }$

${\displaystyle A=[1;3,3,1,3,1,2,1,2,1,4,2,35,21,1,4,4,1,1,3,2,17,7,4,1,3,16,5,3,2,3,1,4,8,1,1,19578,1,1,1,1,1,8,\ldots ]}$(Follow A123561 in OEIS )

If you choose the first five values ​​of this sequence of numbers, you get:

${\displaystyle A\approx [1;3,3,1,3]=1+{\frac {1}{3+{\frac {1}{3+{\frac {1}{1+{\frac {1}{3}}}}}}}}={\frac {64}{49}}\approx 1{,}30612<A}$

If you choose the first six values ​​of this sequence of numbers, you get:

${\displaystyle A\approx [1;3,3,1,3,1]=1+{\frac {1}{3+{\frac {1}{3+{\frac {1}{1+{\frac {1}{3+{\frac {1}{1}}}}}}}}}}={\frac {81}{62}}\approx 1{,}30645>A}$

If you choose the first seven values ​​of this sequence of numbers, you get:

${\displaystyle A\approx [1;3,3,1,3,1,2]=1+{\frac {1}{3+{\frac {1}{3+{\frac {1}{1+{\frac {1}{3+{\frac {1}{1+{\frac {1}{2}}}}}}}}}}}}={\frac {226}{173}}\approx 1{,}30636<A}$

These continued fractions alternately result in excessively large or too small approximate fractions of . The approximate fractions that can be obtained from the above continued fraction expansion are as follows: ${\displaystyle A}$

${\displaystyle {\begin{aligned}&{\frac {1}{1}},{\frac {4}{3}},{\frac {13}{10}},{\frac {17}{13}},{\frac {64}{49}},{\frac {81}{62}},{\frac {226}{173}},{\frac {307}{235}},{\frac {840}{643}},{\frac {1147}{878}},{\frac {5428}{4155}},{\frac {12003}{9188}},{\frac {425533}{325735}},{\frac {8948196}{6849623}},{\frac {9373729}{7175358}},\\&,{\frac {46443112}{35551055}},{\frac {195146177}{149379578}},{\frac {241589289}{184930633}},{\frac {436735466}{334310211}},{\frac {1551795687}{1187861266}},{\frac {3540326840}{2710032743}},\ldots \end{aligned}}}$

Generalizations

• There is no reason why the middle part of the double exponential function above has to be a 3. In fact, L. Kuipers and AR Ansari were able to generalize this finding by showing that: ${\displaystyle \lfloor A^{3^{n}}\rfloor }$

There is every real number , a constant , so is prime for all positive integers . ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle c\geq 2{,}106}$ ${\displaystyle A}$ ${\displaystyle \lfloor A^{c^{n}}\rfloor }$ ${\displaystyle n\in \mathbb {N} }$

• You can also replace the rounding function ( ) with the rounding function ( ). The mathematician László Tóth was able to prove the following statement in 2017: ${\displaystyle \lfloor \cdot \rfloor }$ ${\displaystyle \lceil \cdot \rceil }$

There is every natural number , a constant , so is prime for all positive integers . ${\displaystyle r\in \mathbb {N} }$ ${\displaystyle r\geq 3}$ ${\displaystyle B}$ ${\displaystyle \lceil B^{r^{n}}\rceil }$ ${\displaystyle n\in \mathbb {N} }$

Example: Be ${\displaystyle r=3}$

Then (follow A300753 in OEIS ) ${\displaystyle B=1{,}24055470525201424067\ldots }$

The prime numbers thus generated are:

2, 7, 337, 38272739, 56062005704198360319209, 176199995814327287356671209104585864397055039072110696028654438846269, ... ( continuation A118910 in OEIS )

See also

• Prime number generator

Web links

• Eric W. Weisstein : Mills' Constant . In: MathWorld (English).
• E. Kowalski: Who remembers the Mills number? (English)
• Awesome Prime Number Constant , Number Phile (English)

Individual evidence

1. ^ William H. Mills: A prime-representing function . In: Bulletin of the American Mathematical Society . tape 53 , no. 6 , 1947, ISSN  0002-9904 , pp. 604 ff ., doi : 10.1090 / S0002-9904-1947-08849-2 .
2. ↑ ((((((2521008887 ³ + 80) ³ + 12) ³ + 450) ³ + 894) ³ + 3636) ³ + 70756) ³ + 97220 on Prime Pages
3. ↑ List of the 5000 largest known prime numbers (English). Retrieved December 23, 2019 . 
4. ↑ Henri Lifchitz, Renaud Lifchitz: PRP Top Records - Search by form (((((((((? +450) ^ 3 +? PRP Records, accessed on January 2, 2020) . 
5. ↑ Chris K. Caldwell, Yuanyou Cheng: Determining Mills' Constant and a Note on Honaker's Problem . In: Journal of Integer Sequences . Vol. 8, No. 4 , 2005 ( full text ).
6. ↑ Steven R. Finch: Mills' Constant . In: Mathematical Constants . Cambridge University Press, 2003, ISBN 0-521-81805-2 , pp. 130-133 .
7. ^ ^{A b} László Tóth: A Variation on Mills-Like Prime-Representing Functions. Journal of Integer Sequences, Vol. 20 , Article 17.9.8, 2017, pp. 1–5 , accessed on January 2, 2020 .