Coprimality of consecutive Euler up/down numbers and the preperiod at odd prime powers

Abstract

Let Eₙ be the number of alternating permutations of {1,…,n}, so that sec z+tan z=Σ_(n ≥ 0)Eₙzⁿ/n!. Knuth and Buckholtz proved that (Eₙ mod q) is eventually periodic; write d(q) for its minimal eventual period and s(q) for its preperiod. Ramassamy conjectured s(pʳ)=r at every odd prime power; Güleç has recently disproved this with s(5⁵)=4, proved the companion period conjecture, and — on the strength of a search over pʳ ≤ 10¹⁰⁰⁰⁰⁰ — conjectured the weaker bound s(pʳ) ≥ r-2. We observe that his divisibility criterion turns that conjecture into a statement about greatest common divisors of consecutive Euler numbers: the single integer gcd(Eᵣ₋₃,Eᵣ₋₂,Eᵣ₋₁) settles the index r for every prime p at once, with no upper bound on p whatsoever. Computing the Euler numbers exactly to index 10 000 we prove that no prime divides three consecutive Euler up/down numbers of index at most 10 000, hence that s(pʳ) ≥ r-2 for every odd prime p and every 4 ≤ r ≤ 10001 — a region no search organised by the size of pʳ can cover. We classify, over all odd primes at once, the cells with s(pʳ) ≤ r-2 in that range: they are exactly p=43 and r ∈ {980,2786,4592,6398,8204}. In particular 43⁹⁸⁰ is the smallest odd prime power with s(pʳ) ≤ r-2, a minimality the source only suggests. We also determine the primes dividing two consecutive Euler up/down numbers below index 10 000 — exactly 43, 433, 1093, 5107, 13367 — and reproduce independently every computational assertion of the source's final section that names an explicit value, assertions its own declaration attributes to language models and for which no code is distributed. Every computation below is machine-checked in Lean 4; the passage from divisibility to preperiods is the source's criterion, quoted and not reproved.

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