More information about Moser formula for primes: $ p_n=\gamma \cdot 10^{n(n+1) / 2}-10^n \gamma \cdot 10^{n(n-1) / 2} $

Question

You will find the above formula here.

The problem is I can't find any deduction of the formula above anywhere.

I couldn't find the reference number 61 "Leo Moser, A prime representing function, Mathematics Magazine 23:3 (1950), pp. 163–164" on the internet.

I only have found another expression for Moser's formula in a spanish article Fórmulas que generan números primos: $$ p_n=A\left(10^{n(n+1) / 2}-10^n A \cdot 10^{n(n-1) / 2}\right) $$

$$A= 0.203005000700011...$$

Thanks in advance for any information about the above constant.

Answer 1

I can't give you a reference, but it looks like Moser has written:

$$\lambda =\sum_{n=1}^{\infty}\frac{p_n}{10^{n(n+1)/2}}$$ and thus encoded the primes in the decimal digits.

Then defining:

$$D_n= 10^{n(n+1)/2}\lambda\\ C_n=\lfloor D_n\rfloor$$

We have: $$\begin{align} C_n&=p_n+10^{n}p_{n-1}+10^{n+n-1}p_{n-2}+\cdots +10^{n+(n-1)+\cdots+2}p_1\\&= p_n+10^nC_{n-1} \end{align} $$

So $p_n=C_n-10^nC_{n-1}.$

Without the floor functions, the expression is, of course, zero. We can replace the floor function with integer rounding, which will amount to the same thing since $2p_n<10^n.$

We could do this for any sequence of positive integers with $a_n<10^n-1.$ We'd get a $\lambda$ which encodes $\{a_n\}.$

This is one of those things which is showing how much information can be stuffed into a real number. It really shows nothing about the primes.

There is nothing particularly important about $f(n)=10^{n(n+1)/2},$ other than the fact that $f(n)/f(n-1)$ is always an integer and $f(0)=1$ and $p_n+1<\frac{f(n)}{f(n-1)}.$ Under those circumstances, if we define $\lambda =\sum_n \frac{p_n}{f(n)},$ we get:

$$p_n=\lfloor f(n)\lambda\rfloor -\frac{f(n)}{f(n-1)}\lfloor f(n-1)\lambda\rfloor$$