We study the Diophantine equations (k!)n − k n = (n!)k − n k and (k!)n + k n = (n!)k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or (k, n) = (1, 2), (2, 1) and that the second one holds if and only if k = n.
A photometric membership test has been carrled out for stars in the region of the open cluster M 39 (= NGC 7092 = OCL 211 = C 2130 + 482) selected by ARTIUKHINA, KALININA from proper motions to be probable members of the cluster corona. Using own UBV observations of 28 brighter (BD) stars and additional photometric data from other available sources out of the 117 probable members in the region outside of the cluster core altogether for 37 the membership could be confirmed. 22 of the stars do not belong to the cluster. The results confirm the existence of the cluster corona found statistically by ARTIUKHINA. The distribution of the confirmed members over the investigated area of about 5' diameter indicate that the corona of this poor cluster is much more extended than 75' as suggested by the statistical investigation.
In recent papers, S. N. Begum and A. S. A. Noor have studied join partial semilattices (JP-semilattices) defined as meet semilattices with an additional partial operation (join) satisfying certain axioms. We show why their axiom system is too weak to be a satisfactory basis for the authors’ constructions and proofs, and suggest an additional axiom for these algebras. We also briefly compare axioms of JP-semilattices with those of nearlattices, another kind of meet semilattices with a partial join operation.