Roczniki Filozoficzne

Volume 19, Issue 3, 1971

On Numbers Satisfying the Congruence

In the present paper the problem of finding numbers x of n-digits satisfying the congruence: x2 — kx = 0 (mod.10n) was solved; where k = integer, n = 1, 2, 3... and even a more general congruence x2 — kx = 0 (mod. gn). Three cases were examined: 1) k = 1; 2) k= -1; 3) |k|>1. Thus the paper consists of three parts. In part I the first case k = 1 was analysed. This problem in such a form was first introduced by P. Tedenat in 1814. It was solved in this paper by means of three methods: a) the method of recurring sequences, b) the method of continued fractions, c) the method of .power sequences. The author also gives ways to calculate x in the case of any numbers of digits; effectively numbers of x are given for n = 21, whereas E. Lucas gave them only for n = 10. In part II the author analyses the so far unexamined case no. 2: k= -1. The above mentioned methods a), b), c) all apply to this case. Case no. 2 is the parallel case no. 1. In part III the author examines case no. 3, which has not been also treated by other authors, namely | k | > 1. It represents a generalization of the two former cases and can be easily reduced to them by means of a substitution. Laisit of ail the author generalizes the given problem for the congruence x2 — ax + c=0 (mod. 10n) in the case, when A = a2 — 4c = m2 and m = integer.