Irrationality of the finite analogue e_A

Ascertain whether e_A ∈ A, defined by e_A = D_A(0) = (∑_{k=0}^{p-1} 1/k! mod p)_p where A = (∏_{p prime} Z/pZ)/(⊕_{p prime} Z/pZ), is irrational in A; that is, determine whether e_A lies outside the embedded copy of Q in A.

Background

Motivated by Dobiński’s formula for e, the authors define a finite analogue e_A via truncated factorial sums reduced modulo p and aggregated in A. They prove an A-analogue of Dobiński’s formula decomposing D_A(n) in terms of e_A and a recursively defined sequence.

Despite these structural results, the arithmetic nature of e_A remains unclear. In contrast with the classical setting where e is transcendental, the authors emphasize that even irrationality of e_A (non-membership in Q ⊂ A) has not been established.

References

However, it remains unknown whether $e_A$ is irrational in the ring $A$, or whether $D_{r,A}(j)$ for $0 \le j \le r-1$ are $Q$-linearly independent.

— On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials  (2604.01578 - Matsusaka et al., 2 Apr 2026) in Section 1 (Introduction)