A Generalised Form of a Theorem on Integer Quotients of Products of Factorials

V1 V'2 PI P2 ... where v1, v2, . . . are ? 0 of which all but a finite number are 0. Call these numbers integers and suppose that no two generalised are equal if their v's are different. Then arrange {1} as an increasing sequence: I = 11 < 12 < 13 < . . . < In <* L. J. Mordell, [2], has written on Integer quotients of products of factorials. The aim of this note is to prove three theorems on this topic for generalised integers. Define In= I i , and [I] the number of I numbers < 1, e.g., [In] = n THEOREM 1. If [In] = [la] + [lb] + [le] + * * * [1k] the quotient In ! (1) la Vlb 1lC ! . . . Ik is a generalised integer. Proof. In [1], I have proved that the exponent of the highest power of a generalised prime P which divides In! is 8In