Necessary conditions on maximal intervals of consecutive composites

For N >= 2, Co(N) denotes the number of consecutive composites immediately following N. For a composite N, a divisor d is a larger factor if d >= N/d, equivalently d^2 >= N; the least of them, D(N), is the minimum of max{m,n} over all decompositions N = mn with m, n >= 2. The inequality Co(N) < D(N) is thus equivalent to Co(mn) < max{m,n} for every such decomposition. The main theorem shows that this inequality, holding throughout a maximal interval of consecutive composites, forces the larger factors of its elements to be pairwise distinct: no two elements of the interval share a larger factor. Two criteria for locating primes follow. First: if A < B share a divisor d that is a larger factor of A, and the inequality holds at A, then [A,B] contains a prime. Second, requiring no information about the endpoints: if [A,B] encloses two consecutive terms of the sequence floor(n^2/4) and the inequality holds at the first of them, then [A,B] contains a prime. Two further restrictions of the same kind are obtained. All arguments are elementary and self-contained. Part of independent research in elementary number theory.