Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1978. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 14)

IRODALOM [1[ Annales de Math., 5 (1814-15). [2] R. D. Carmichael, Note on a new number theory funcion, Bull of the Amer. Math. Soc., 16 (1910), 232-238. (3| A. Cunningham: On agreeable numbers, British Assoc. Report, 1893, 699. |4] L. E. Dickson, History of the theory of numbers, Chelsea Publ. Co., New York, 1971, vol. I. [5] Forhandlinger i videnskabs-selsk. i Christiania, 1901 (Oversigt over Selsk. Moder; 1901), 3-13. [6] R. L. Goodstein, Numbers in a general scale, Math. Gaz., 43 (1959), 270-272. [7] E. Hewitt, Certain congruences that hold identically, Amer. Math. Monthly, 83 (1976), 270-271. [8] P. Kiss, On one way of making automorphic numbers, Publ. Math. Debrecen, 22 (1975), 199-203. [91 P. Kiss, Aufabe 768, Elemente der Math., 31 (1976), 72. (10) N. I. Nedita, O probléma de teória numerelos, Gazeta Matematica, ser. A, 73 (1968), 191-196. [11| Nicholas P. Callas, Representations of automorphic numbers, Fibonacci Quart., 10 (1972), 393-396,402. [12| C. P. Popovici, Une généralisation d'une equation arithmétique de D. Pompeiu, Bull. Math, de la Soc. Sei. Math, de Roumania, Tome 13 (61), 1969, 73-84. 113] A. Rotkiewicz. Pseudoprime numbers and their generalizations, University of Novi Sad, 1972. [141 Vernon de Guerre and R. A. Fairbairn, Automorphic numbers, Journal of Reer. Math., Vol. 1 (1968), 173- 179. ON A BINOM CONGRUENCE BY PÉTER KISS In this paper we solve a generalization of a classical problem. The problem was drawn first up in the Annales de Math. ([1], p. 220): What is the number of which successive powers end in the same number of n digits as the originál number? This problem leads to the solution of the congruence x

2-x —O (mod 10

n). We solve the congruence x^-x — 0 (mod B

n), where k, B and n are fixed integers, and we give the explicit form of the solutions and the number of solutions. We show that some results of N. I. Nedita (101, C. P. Popivici (12], E. Hewitt [7]andR. D. Carmichael [2[ follow from our results. We solve a problem of K. Szymiczek (see [13], problem 46, p. 143) concerning pseudoprime numbers, as an application of our results. 464