Commute.pow_dvd_pow_of_sub_pow_eq_zero

theorem Commute.pow_dvd_pow_of_sub_pow_eq_zero : ∀ {R : Type u_1} {x y : R} {n m p : ℕ}, n + m ≤ p + 1 → ∀ [inst : Ring R], Commute x y → (x - y) ^ n = 0 → x ^ m ∣ y ^ p

The theorem `Commute.pow_dvd_pow_of_sub_pow_eq_zero` states that for any given ring `R` and elements `x`, `y` of `R`, and any natural numbers `n`, `m`, `p` such that `n + m` is less than or equal to `p + 1`, if `x` and `y` commute (i.e., `x * y = y * x`) and `(x - y) ^ n` equals zero, then `x ^ m` divides `y ^ p`. In terms of number theory, it means that if two numbers commute and the `n`-th power of their difference is zero, then the `m`-th power of the first number is a divisor of the `p`-th power of the second number, given that the sum of `n` and `m` does not exceed `p + 1`.

More concisely: If `x` and `y` are commuting elements in a ring, and `(x - y)^n = 0` for some natural number `n`, then `x^m` divides `y^(n+m)` for any natural number `m` with `n+m <= p+1`.