Erdos Problem No. 686 ˝ Partial Results on the Square Cases: the Example 23782 , Runge's Method for k = 8 and k = 12, and a Prime Number Lemma

Erdos Problem No. 686 asks whether every integer ˝ N ≥ 2 can be written as ∏ki=1(m+i)∏ki=1(n+i) withk ≥ 2 and m ≥ n + k. We record partial results on the open square cases. (i) 23782is representable withk = 4, via a Pell equation. (ii) For k = 8, Runge’s method gives the effective bound (2n+9)2 < 4096t +42for any representation of t2; combined with an exact finite search, 4 and 25 are not 8-representable. (iii) Fork = 12, we prove a uniform sign threshold and the effective bound (2n+13)2 < 2223936t +143; an exactsearch then shows that 4 and 25 are not 12-representable. (iv) We characterize when Runge’s remainderRk is a perfect square (a Prouhet–Tarry–Escott-type partition condition), verify that this happens only fork ∈ {2,4,8} among even k ≤ 30, and explain why the method gives no bound uniform in k. (v) We provethat any representation forces the interval [m+1,m+k] to contain no prime > N, linking the case N = 4 togaps between consecutive primes. All computations use exact integer arithmetic; the scripts are included.