Elementary Properties of Interval Lengths of Consecutive Primes

We study the interval lengths determined by consecutive primes. For integers k ≥ 2 and 1 ≤ m < k, define S_m(k) = p_k − p_(k−m), the length of the interval containing the m + 1 consecutive primes p_(k−m), ..., p_k. We also introduce the associated extremal statistic κ_k = max{0 ≤ m ≤ k − 1 : S_m(k) ≤ g_k}, where g_k = p_(k+1) − p_k and S_0(k) = 0 by convention. Two elementary results are established. First, the maximal value κ_k = k − 1 occurs if and only if k = 2. The proof relies on a classical form of the Bertrand–Chebyshev theorem. Second, the sharp lower bound S_m(k) ≥ 2m holds for all k ≥ 3 and 1 ≤ m < k. Equality is completely classified: it holds for m = 1 precisely when the preceding gap is 2; it holds for m = 2 only in the single case corresponding to the primes 3, 5, 7; and it never holds for any m ≥ 3. These results provide a structural characterization of interval spans determined by consecutive primes and clarify their relationship with the immediately following prime gap.