The Hardy–Littlewood Second Conjecture (HL₂): A Theorem on the Non-Increase of Primes under Translation

This paper employs elementary sieve methods, together with the periodic skeleton structure modulo 30 and a disjoint partition of composite numbers by their least prime factor, to compare the prime counts in two intervals of equal length, [1, y] and (x, x+y]. The goal is to prove the Hardy–Littlewood Second Conjecture (HL₂): for integers x, y ≥ 2, π(x+y) ≤ π(x) + π(y), which is equivalently expressed as π(y) ≥ π(x+y) − π(x), that is, the number of primes in any translated interval (x, x+y] of length y does not exceed the number of primes in the initial interval [1, y]. The proof is supported by three components: reduction via interval overlap, verification on finite intervals, and qualitative analysis on large intervals. First, the initial interval starts at 1 and therefore has a higher count of odd numbers; a translated sliding interval may land at positions where the proportion of odd numbers is lower, making the relative proportion of composites more likely to be higher. Removing even numbers is an operation applied synchronously to both intervals and does not change the relative strength of their prime counts. Second, primes greater than 5 can only lie in the eight admissible residue classes modulo 30 — 1, 7, 11, 13, 17, 19, 23, 29 — forming a periodic skeleton modulo 30. Any nine consecutive odd numbers can contain at most four primes, namely the 4/9 high-density configuration. Once the interval length exceeds one period-30 skeleton, the local prime-clustering effect is diluted by the periodic averaging effect, so scattered high-density fragments cannot raise the overall prime density above that of the initial interval. Third, taking primes not exceeding y as sieves, the numbers of multiples of the same prime p in the two equal-length intervals differ by at most 1; the initial interval contains these small primes themselves, yielding a small-prime bonus that cancels this endpoint-induced counting deviation. Fourth, composites are partitioned disjointly by their least prime factor, so that each composite uniquely belongs to the sieve set of its least prime factor, naturally avoiding the alternating perturbations of inclusion–exclusion. Within the initial interval, many multiples of a prime p have already been sieved out by smaller primes and are not counted again in p's contribution, stabilising the counting baseline. Fifth, a sliding interval with larger numerical values activates more new sieves, producing new composites absent from the initial interval, and further reducing the number of primes retained in the sliding interval. The whole argument uses only elementary integer properties, modular periodic structure, and least-prime-factor counting, without relying on advanced analytic number theory tools such as complex analysis or elliptic curves.