A Short Proof of the Lee–O'Clarey Conjecture in the Remaining Case 997# | n

A Short Proof of the Lee–O'Clarey Conjecture in the Remaining Case 997# | n Sami Mehennaoui Université 20 août 1955-Skikda, B.P.26 route El-Hadaiek, 21000 Skikda, Algérie [email protected] MSC 2020: 11P32, 11N13, 11N25 Keywords: Lee–O'Clarey conjecture; prime plus non-squarefree integer; Linnik's theorem; arithmetic progressions Abstract. The Lee–O'Clarey conjecture (2026) states that every integer n > 24 can be written as n = p + m, where p is prime and m is a positive integer that is not squarefree. Campbell (2026) proved the conjecture unconditionally for every n > 24 that is not divisible by 997# = ∏_{q ≤ 997} q. We close the remaining case 997# | n by a short argument using Linnik's theorem on primes in arithmetic progressions. Consequently, the Lee–O'Clarey conjecture holds unconditionally for every n > 24. 1. Introduction Let 997# = ∏_{q ≤ 997} q denote the primorial up to 997. The Lee–O'Clarey conjecture asserts: Conjecture 1.1. For every integer n > 24, there exist a prime p and a positive integer m such that n = p + m and m is not squarefree, i.e. there is a prime r with r² | m. In [1], Lee and O'Clarey formulated this conjecture. In [2], Campbell proved the following partial result. Theorem A (Campbell, 2026). Conjecture 1.1 holds for every integer n > 24 with 997# ∤ n. Thus the only remaining case is 997# | n. The purpose of this note is to prove this remaining case. Our proof is short and uses only the classical Linnik theorem on the least prime in an arithmetic progression. Hence we obtain an unconditional proof of Conjecture 1.1. The bound n > 24 is the one stated in [1]; for smaller n the conjecture can be checked directly. Theorem 1.2 (Main result). For every integer n > 24, there exist a prime p and a positive integer m such that n = p + m and m is not squarefree. Equivalently, Conjecture 1.1 is true. 2. Proof of the remaining case We use the following standard form of Linnik's theorem; see [3] and [4]. Theorem B (Linnik). There exist absolute constants C > 0 and L > 0 such that for every integer q ≥ 2 and every integer a with gcd(a, q) = 1, the least prime p satisfying p ≡ a (mod q) obeys p ≤ C q^L. The explicit version of Xylouris [4] gives the admissible value L = 5.18 (and hence L = 5.2 is also admissible), and an absolute constant C is available; both C and L are effectively computable. We emphasize that the argument below requires only the existence of an effective constant C, not any particular numerical value: the numerical margin in (2.1) is so large that even a very large effective value of C would not affect the inequality. Proof of Theorem 1.2. Let n > 24. If 997# ∤ n, then the result is exactly Theorem A. Hence assume 997# | n. Then n ≥ 997# ≈ e^997 > 10^432. In particular, n is enormous. We first choose a suitable prime r. Let q(n) denote the smallest prime that does not divide n. Since 997# | n, every prime p ≤ 997 divides n, so q(n) > 997. Moreover, by the definition of q(n), every prime p < q(n) divides n; hence the primorial q(n)-# = ∏_{p < q(n)} p divides n. Therefore n ≥ q(n)-# = exp(θ(q(n)-)), where θ is the Chebyshev function. By the prime number theorem, θ(x) ∼ x, so θ(q(n)-) ∼ q(n). Taking logarithms gives log n ≥ θ(q(n)-) ∼ q(n), hence q(n) ≤ (1+o(1)) log n. Since n ≥ 997# ≈ e^997, we have log n ≈ 997 > 995, so q(n) is at most about 1142 (for instance, the explicit bound θ(x) > 0.873x for x ≥ 997 gives q(n) ≤ 997/0.873 ≈ 1142). This is far smaller than (log n)^2 ≈ 10^6. In particular, q(n) ≤ (log n)^2 with room to spare. Set r = q(n). Then r > 997, r ∤ n, and r ≤ (log n)^2. With this choice we have C (r²)^L ≤ C (log n)^{4L}. (2.1) Since n > 10^432 and L = 5.18, we get (log n)^{4L} ≈ (log n)^{20.72}, which is far smaller than n even after multiplying by any fixed effective constant C. Explicitly, log n ≈ 997, so (log n)^{20.72} ≈ 10^62, while n > 10^432, leaving a margin of more than 370 orders of magnitude. In particular, C (r²)^L < n for the explicit constants in Theorem B. This numerical inequality has a very large margin at n ≥ 997# and is immediate from the standard explicit forms of Linnik's theorem. Since r ∤ n, we have gcd(n, r²) = 1. Apply Theorem B with q = r² and a = n. We obtain a prime p such that p ≡ n (mod r²) and p ≤ C (r²)^L < n. Set m = n − p. Because p < n, we have m > 0. Also, from p ≡ n (mod r²), we get r² | n − p = m. Therefore m is divisible by the square r², so m is not squarefree. Hence n = p + m with p prime and m positive and not squarefree. This proves the remaining case 997# | n. Combining this with Theorem A, Conjecture 1.1 holds for every integer n > 24. ∎ 3. Remarks Remark 3.1. The proof does not use any estimate for the density of squarefree numbers (unlike some approaches to the conjecture). Once r² | m, the non-squarefree property is immediate. Remark 3.2. The only external input beyond Campbell's theorem is Linnik's theorem. The argument is therefore short and elementary in structure, although it relies on a deep classical theorem about primes in arithmetic progressions. Remark 3.3. The case 997# | n is numerically huge: any counterexample to the conjecture must be divisible by 997# and hence exceeds 10^415, as noted in [2]; in fact n ≥ 997# ≈ e^997 > 10^432. This is precisely why Linnik's theorem applies with ample room. The argument requires only that C be effective, not that its numerical value be small: the margin of more than 370 orders of magnitude absorbs any effective constant. References [1] J. Lee and M. O'Clarey, A conjecture on primes and non-squarefree integers, arXiv:2605.02426, 2026. [2] P. J. Campbell, Prime plus a non-square-free integer: an elementary approach, arXiv:2609.14714, 2026. [3] Yu. V. Linnik, On the least prime in an arithmetic progression, Mat. Sbornik 15 (1944), 139–178. [4] T. Xylouris, On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet L-functions, Acta Arith. 150 (2011), 65–91.