English
In a commutative semiring, for prime p and elements u, v with v | u and p u v | u^p, there exists y such that (1 + u x)^(p^m) = 1 + p^m u (x + v y) for any x, m.
Русский
В коммутативном полускольном кольце для простого p и элементов u, v с v | u и p u v | u^p существует y such that (1 + u x)^(p^m) = 1 + p^m u (x + v y) для любого x, m.
LaTeX
$$$\\\\forall R [CommSemiring R], \\\\forall u, v \\\\in R, \\\\forall p \\\\in\\\\mathbb{N}, \\\\text{Prime}(p) \\\\Rightarrow \\\\forall x \\\\in R, \\\\forall m \\\\in \\\\mathbb{N}, \\\\exists y \\\\in R, (1 + u x)^{p^{m}} = 1 + p^{m} u (x + v y).$$$$
Lean4
theorem orderOf_one_add_prime {p : ℕ} (hp : p.Prime) (hp2 : p ≠ 2) (n : ℕ) :
orderOf (1 + p : ZMod (p ^ (n + 1))) = p ^ n :=
by
convert orderOf_one_add_mul_prime hp hp2 1 _ n
· simp
· intro H
apply hp.ne_one
simpa using Int.eq_one_of_dvd_one (Int.natCast_nonneg p) H
