Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria

A superseded deduction

Factorial-congruence multiplicities give another lower bound for $L_N$. Its exponent is weaker than that of the terminal-block argument in §3. We include the deduction to specify what follows from this external input. For an odd prime $r$, let $m_r$ count the indices $2\le n\le N$ with $r\mid n!-1$; such an index satisfies $n<r$. The factorial-congruence multiplicity estimate of Garaev, Luca and Shparlinski [7], applied to the residue $a=1$ on $1\le n\le min(N,r-1)$, gives $m_r\ll N^{2/3}$; the prime $2$ divides none of these factors. The sum of the exponents at a prime is at most the number of occurrences times the largest exponent, which is its exponent in $L_N$. Thus

and Stirling’s formula makes the left side $\asymp N^2\log N$, whence $\log L_N\gg N^{4/3}\log N$. Theorem 3.4 supersedes this: its exponent is $3/2$, its constant is explicit, and it does not use the external multiplicity theorem. The deduction is retained because it is the only place a multiplicity bound enters the record.

Dividing all coefficients by their gcd does not evade the lcm restriction: the resulting integer vector still has $V_2=\cdots =V_D=0$, so its value of $M$ is still divisible by $L_D$. For the nonzero cofactor vector, §1.5 justifies normalisation by identifying the primitive progression vector. It supplies no proof that the remainder is nonintegral. No formalised primitive cofactor construction is claimed here.