Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria

A non-polynomial comparison.

For $n!+2^n-1$, Luca and Shparlinski [19] eliminate the exponential term using three indices satisfying the congruence; multiplicative order enters the counting. Their valuation-layer identity [19] records repeated prime-power divisibility. These are related tools, not inputs to Theorem 3.4, and no extension to general non-polynomial perturbations is asserted.

Within the terminal-block estimate just proved, maximising $\alpha -{\alpha }^3/6$ gives $2\sqrt{2}/3$. This is an optimisation of that one-parameter bound, not an optimality theorem for all block selections or all gcd arguments. The discarded factorial term has size $N^{3/2}$, so a finite normalised ratio need not be close to the limiting lower bound.

Theorem 3.4 concerns $L_N$, not $\mathrm{den}(H_N)$. Theorem 2.1 and the cancellations at $139$ and $2593$ show why these cannot be interchanged: a prime present in the common denominator may be absent after reduction. The two examples establish this possibility, not an asymptotic estimate for cancellation.

Even two consecutive denominators obstruct clearing by the full lcm. For $N\ge 3$, the gcd of $(N-1)!-1$ and $N!-1$ divides $N-1$, whereas $(N-1)!-1$ is coprime to $N-1$. Thus those denominators are coprime, and

This argument does not use the stronger asymptotic growth theorem.

Reduction could still help, but only in the opposite direction: a clearing argument needs a sufficiently small reduced denominator. Write $H_N={\hat{A}}_N/Q_N$ in lowest terms, reserving $A_N$ for the numerator over the full lcm. Under $S=a/q$, with $a\in \mathbb{Z}$ and $q\ge 1$, positivity gives

A contradiction requires $q{Q}_N(S-H_N)<1$ at some $N$. A lower bound on $Q_N$ cannot supply this upper estimate. For comparison, the rational limit $1$ and the approximants $1-1/Q_N$ have arbitrarily large reduced denominators without any irrationality. The inequality $d_N(S-H_N)<1$ alone would not repair the argument, because $d_N$ generally does not clear the earlier summands of $H_N$. The prime-power analysis describes which factors survive reduction, but it does not yet supply the required upper control on $Q_N$.

For linear forms in $p$-adic zeta values, with $p\ge 5$ a fixed prime and $n$ the index of the form, Lai, Lupu and Sprang quantify a saving of this kind. They bound the denominators of the coefficients by powers of $\mathrm{lcm}(1,\dots ,n)$, show that the bound may be divided by a product ${\mathrm{\Phi }}_n$ of prime powers, and compute the growth rate of ${\mathrm{\Phi }}_n$ [15]. That saving enters the inequality [15] under which the rescaled forms, which have integer coefficients, meet the irrationality criterion [15] that they quote from Lai [16]. Along one unbounded subsequence, the forms must be nonzero and their $p$-adic absolute values, multiplied by the largest ordinary absolute value of an integer coefficient, must tend to zero. The coefficient bound is archimedean; the bound on the value is $p$-adic. The conclusion is that at least one of the finitely many $p$-adic numbers in the forms is irrational, not that each is irrational. The corresponding saving for $H_N$ is the ratio $L_N/Q_N$.

This ratio is an integer. Merely naming it supplies neither its growth nor a small linear form; the analogy is about what an eventual estimate would have to accomplish. Theorem 2.1 decides, for each prime $p\mid L_N$, whether $p$ divides this ratio; no asymptotic lower bound for the ratio is established here. This paragraph compares the two methods; no $p$-adic theorem is applied to $S$.