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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:

1. If $$b = 0$$, then $$a \uparrow \uparrow_{\beta} b := \beta$$.
2. 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 $$n$$.[1][2] 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 $$n=4$$.[2]

## 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 $$1.92878$$.[2]

## Primes

Here is a list of the first few terms in the sequence of Wright's primes:[2]

• 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 $$\alpha$$.[3]

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.[2] Lowell Schoenfeld showed a related result for a smaller base larger than 1.[4] The existence for the case $$B = 10$$ is referred to in Japanese cartoon ワヘイヘイの日常.[5]

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 $$1.92878$$.

4. 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.