Let , , the class number of . Let be an odd prime number, the exact power of 2 dividing or according as or not, . The authors prove that does not divide for any integer . If , then does not divide for all .
For Part I, see Exp. Math. 18, No. 2, 213–222 (2009; Zbl 1189.11033), Part III, Int. J. Number Theory 7, No. 6, 1627–1635 (2011; Zbl 1226.11119).