Wright's primes

Wright's primes are certain sequences of prime numbers that have the property that the enumeration sequence of them grows as fast as the tetration function. Define \(a \uparrow \uparrow_{\beta} b\), which is a positive real number defined for non-negative integers \(a\) and \(b\) and real \(\beta\) in the following recursive way:
# If \(b = 0\), then \(a \uparrow \uparrow_{\beta} b := \beta\).
# If \(b \neq 0\), then \(a \uparrow \uparrow_{\beta} b := a^{a \uparrow \uparrow_{\beta} (b-1)}\).
Wright proved that there exists a real number \(\alpha\) such that \(\lfloor 2\uparrow\uparrow_\alpha n\rfloor\) is prime for all natural \(n\).E. M. Wright (1951). "A prime-representing function". American Mathematical Monthly. 58 (9): 616-618.Robert Baillie, WRIGHT'S FOURTH PRIME, v4, arXiv. Let \(\alpha\) denote the smallest positive real number satisfying that \(\alpha \geq 1.92878\) and \(\lfloor 2 \uparrow \uparrow_{\alpha} n \rfloor\) is a prime number for any positive integer \(n\). Then Wright's primes are defined as prime numbers in the sequence \((\lfloor 2 \uparrow \uparrow_{\alpha} n \rfloor)_{n=1}^{\infty}\). Note that the constant \(1.92878\) is Wright's example for producing three primes, i.e. \(\lfloor 2 \uparrow \uparrow_{1.92878} n \rfloor\) is a prime number for \(n = 1,2,3\). However, Wright's example outputs a composite number for \(n=4\).
== Existence ==
For any fixed real number \(N\), the existence of a positive real number \(a\) satisfying \(a \geq N\) and \(\lfloor 2 \uparrow \uparrow_a n \rfloor\) is a prime number for any positive integer \(n\) follows from Bertrand's postulate and the completeness of the real numbers. The existence of \(\alpha\), i.e. the minimum of such an \(a\) for the case \(N = 1.92878\), follows from the fact that the map \(a \mapsto \lfloor 2 \uparrow \uparrow_a n \rfloor\) commutes with the limit for any descending sequence for any natural number \(n\).
== Decimal expansion ==
The decimal expansion of the constant \(\alpha\) is given as
\begin{eqnarray*}
1.92878 \underbrace{0000000000 \cdots 0000000000}_{4927} 82843 \cdots,
\end{eqnarray*}
and hence is approximated to Wright's original constant \(1.92878\).
== Primes ==
Here is a list of the first few terms in the sequence of Wright's primes:
* 3
* 13
* 16381
* 191396642046311049840383730258...303277517800273822015417418499 (4932 digits)
== Variants ==
The original study of Wright's primes by Robert Baillie considers the minimality of \(\alpha\) greater than or equal to Wright's original constant \(1.92878\) generating a sequence of primes numbers whose first three terms match Wright's original sequence 3, 13, 16381 corresponding to \(1.92878\). Replacing the generating condition, we obtain other sequences of primes generated in a similar way.
Charles Greathouse defined a sequence \((a_n)_{n=0}^{\infty}\) of primes such that \(a_0 = 3\) and \(a_{n+1}\) is the greatest prime smaller than \(2^{a_n+1}\) for any natural number \(n\). Then the first three terms of this sequence also match Wright's original sequence, but the corresponding constant is bigger than the constant \(\alpha\).A016104 in OEIS.
It is known that even when the base \(2\) is replaced by a larger number \(B\), the existence of such a constant, i.e. a positive real number \(\beta\) such that \(\lfloor B \uparrow \uparrow_{\beta} n \rfloor\) is a prime number for any natural number \(n\), follows from a similar proof above. Lowell Schoenfeld showed a related result for a smaller base larger than 1.Lowell Schoenfeld, Sharper bounds for the Chebychev Functions \(\theta(x)\) and \(\psi(x)\). II, Mathematics of Computation, vol. 30, no. 134 (April, 1976) pp. 337-360. The existence for the case \(B = 10\) is referred to in Japanese cartoon ワヘイヘイの日常.ワヘイヘイの日常 in pixiv.
Robert Baillie estimated the smallest value \(1.251647597790463 \ldots\) of such a constant under the original condition except for removing the restriction that it should be greater than or equal to Wright's original constant \(1.92878\).
== See also ==
* Prime numbers
* Numbers in number theory
* Mills' constant
== Sources ==
Category:Numbers
Category:Prime numbers
Category:Tetration level
Category:Tetration
ja:ライトの定理