Claims

../

1937_01_01_erdos_oblath: Erdős and Obláth (Acta Szeged 8 (1937)) prove that n! = x^k ± y^k has no solution with x, y coprime and xy > 1 when k > 2 is not 4, and none with k = 4 for sufficiently large n; refereed.

1973_05_01_pollack_shapiro: Pollack and Shapiro (Comm. Pure Appl. Math. 26 (1973)) are credited with showing that n! = x^4 - 1 has no solution; the monograph's wider report, the whole coprime case k = 4, is unconfirmed; refereed.

2025_04_07_barfield: 10! equals 48^4 minus 36^4, a solution of n! = x^k - y^k with xy > 1 and k = 4, so the equation does have solutions; the bases share the factor 12, outside the coprime case Erdős and Obláth had settled.

2025_09_09_cambie: Cambie's remark, credited in the site's commentary, that n! = x^4 + y^4 has no solution with x, y coprime and xy > 1, by a congruence modulo 8.

---