Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria

How the classical criteria apply here

Let $s_n<a$ be partial sums tending to $a$. Koepf and Schmersau show that $\lfloor n{s}_n\rfloor =\lfloor na\rfloor$ for all sufficiently large $n$ forces $a$ to be irrational [9]. The strict inequality matters: if $a=A/B$ were rational, then every sufficiently large multiple $n$ of $B$ would give $\lfloor n{s}_n\rfloor <na=\lfloor na\rfloor$. Their rational-term version uses a positive integer multiplier $p_n$ and obtains the floor equality from $n{p}_n{s}_n\in {\mathbb{N}}_0$ and the strict tail bound $a-s_n<1/(n{p}_n)$ [9]. Indeed, integrality makes $n{s}_n$ a multiple of $1/p_n$, so its gap to the least strictly larger integer is at least $1/p_n$; the increase $n(a-s_n)<1/p_n$ cannot cross that integer. For the choice they record after (2.1), the least common multiple of the reduced summand denominators, here $p_n=\mathrm{lcm}\{k!-1:2\le k\le n\}$, the last two denominators already obstruct the tail bound. As proved in §3, they are coprime for $n\ge 3$, so

For $n\ge 4$ this gives $n{p}_n>(n+1)!-1$. Hence the first omitted summand $1/((n+1)!-1)$ already exceeds $1/(n{p}_n)$, and this $p_n$ cannot satisfy their tail hypothesis. This rules out the displayed choice of $p_n$, not the criterion itself. In fact, the least positive integer $p_n$ for which $n{p}_n{H}_n$ is integral is exactly

Indeed, writing $H_n$ in lowest terms shows that its denominator must divide $n{p}_n$. With this least choice the required inequality is

Thus reduction of the partial sum, and the common factor of its reduced denominator with $n$, determine the best scale available in this criterion. The eventual inequality above remains unproved here; failure of a larger clearing multiplier does not establish its failure.

Duverney’s Theorem 3.1 includes the quadratic growth assumption $c{u}_n^2\le u_{n+1}\le c^{\prime }{u}_n^2$ for positive constants $c,c^{\prime }$, while $u_{n+1}/u_n^2\to 0$ for $u_n=n!-1$ [10]. His Corollary 3.2, which allows signs $a_n\in \{-1,1\}$, assumes convergence of the signed series $\sum _n(u_{n+1}/u_n^2-1)$ in (3.6) [10]. Its terms tend to $-1$ here, so the series does not converge. Lean checks the ratio limit and also the divergence of the absolute-value series; the signed condition fails already by the term test. To see the ratio limit directly, divide numerator and denominator by $(n!{)}^2$: the numerator tends to zero and the denominator $(1-1/n!{)}^2$ tends to one.

For a strictly increasing sequence $(n_k)$ of positive integers, Erdős proved irrationality of $\sum _{k}1/n_k$ under the following conditions, with a fixed $\epsilon >0$ [3]:

Barreto, Kang, Kim, Kovač and Zhang treat products of consecutive denominators and weighted extensions [12]. For $a_n=n!-1$ and every fixed $\psi >1$,

Thus neither Erdős’s limsup hypothesis, which uses $\psi =2$, nor the growth hypotheses of their Theorems 2 and 3 hold. The Lean proof of this limit treats the special case $\psi =2$; the estimate above also covers the other bases greater than one used in those theorems. Their proof of Theorem 3 uses the classical criterion, which they trace to Fourier’s proof that $e$ is irrational, that a rational sum of nonnegative rationals with infinitely many positive terms admits no integers $D_N$ clearing the denominators of the partial sums with $\underset{N}{lim inf}{D}_N{r}_N=0$, where $r_N$ is the tail after $N$ terms [12]; their Proposition 12 produces such integers under the hypotheses of that theorem [12]. For the present series, even the full summand lcm $L_N$ makes $L_N(S-H_N)$ tend to infinity, as shown in §3. A smaller integer cannot clear every summand. It can nevertheless clear the partial sum after addition: the least positive such integer is $\mathrm{den}(H_N)$, and every other one is a multiple of it. Thus this criterion would require information about cancellation in the reduced partial sum, not merely an improvement from the product to the lcm. More precisely, if $S=a/q$, then

Thus $\underset{N}{lim inf}\mathrm{den}(H_N)(S-H_N)=0$ is a sufficient target for this rationality criterion; the preceding common-denominator estimates do not establish it.

Dividing the recurrence $Z_m=m{Z}_{m-1}+1-b_m$ by $m!$ and telescoping gives the exact finite identity

so the carry defects $1-b_m$ are integer coefficients of a factorial series. Hančl and Tijdeman classify the rational sums with polynomial coefficients [11]; their denominator is the cumulative linear product $\prod _{n\le N}(an+b)$, and the individual number $N!-1$ does not occur. In the factorial case $a=1$, $b=0$, the criterion of Oppenheim that they reproduce [11] says the following: for integer coefficients $c_n$ with $|c_n|<n$ eventually and $\underset{n}{lim inf}|c_n|/n=0$, the sum $\sum _n{c}_n/n!$ is rational exactly when $c_n=0$ eventually. This formulation concerns integer coefficients and factorial denominators, not reciprocal terms with denominator $n!-1$. However, their introduction records a version of Oppenheim’s result without the liminf assumption: under $|c_m|<m-1$ eventually, rationality is equivalent to eventual vanishing [11]. Changing finitely many terms adds a rational number, so eventual hypotheses suffice. Here $2-m\le c_m=1-b_m\le 2$ gives $|c_m|<m-1$ for $m\ge 4$; the classical result therefore applies directly.

For completeness, the following elementary proof specialises that result to these one-sided bounds. For $N\ge 2$, telescoping $\sum _{m>N}(m-1)/m!=1/N!$ gives

The lower bound follows from $c_m\ge 2-m$ and $N!\sum _{m>N}(m-2)/m!=1-N!\sum _{m>N}1/m!<1$. The upper bound follows by comparison with $\sum _{j\ge 1}(N+1{)}^{-j}=1/N$. If $\sum _{m\ge 3}{c}_m/m!$ is rational, the scaled tails are integers for all sufficiently large $N$. They must then be zero, and subtracting consecutive tails gives $c_m=0$ eventually. The converse is immediate. Since $Z_M/M!\to S$, the displayed finite identity recovers Theorem 4.1. This is a self-contained specialisation of the classical rationality criterion, not a strengthening of it. It proves an equivalence, not the nonvanishing needed for irrationality.