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Source. Theorem 4.3 and its proof, Corollary 4.2 and its proof, preprint p. 15. Read on the rendered page. The edition read is identified on the source card.

Statement

Let $m$ be a Gaussian integer and $(b_n)_{n\ge1}$ a sequence of positive integers for which $b_N,b_{N+1},\ldots,b_{4N}$ form an arithmetic progression for infinitely many positive integers $N$ such that $2N-1$ is prime. Assume that $b_n=o(n\log n)$. Then

$$\sum_{n=1}^{\infty}\frac{m^{b_n}}{n!}\notin\mathbb{Q}[i].$$

The case $m=0$ (an observation of this page). The printed statement admits $m=0$, for which the sum is $0$; the proof writes $m^{b_{N+1}-b_N}$ as a quotient $c/d$ with $c$ a Gaussian integer and $d$ a positive integer coprime to $c$, and the theorem is to be read for $m\ne0$, as the abstract (p. 1) has it.

Corollary 4.2 (p. 15). $\pi$ is irrational. The proof supposes $\pi=t/q$ with $t,q\in\mathbb{N}$ and applies the theorem with $m=it$ and $b_n=n$ to $(-1)^q=e^{it}=\sum_{n\ge0}(it)^n/n!$.

Proof pointer

By the proof of Proposition 3.2 (see Theorem 3.2) it suffices that $D_N\ne0$. Identity (13) with $a=1$, $b=0$ shows that $D_N$ is a Gaussian integer divisible by $N!$; since $2N-1$ is prime, every term in its expansion is divisible by $2N-1$ except the one with $k=N$, $n=2N-1$, so $D_N$ is not divisible by $2N-1$ and is nonzero.

Dependencies

Proposition 3.2 and Lemma 2.3 of the same paper.

Bears on

No catalog problem directly.