CONCERNING A PROBLEM OF H. SCHOLZ By ANDRZEJ MOSTOWSKI in Warsxawa G . ASSER in his recent article [l] has established a number of interesting results pertaining to a problem proposed by H. SCHOLZ [2]. The results of ASSER overlap in part with results which I have found in 1953 while attempting (unsuccessfully) to solve SCHOLR'S problem (cf. Roczniki Polskiego Towarzystwa Matematycznego, series I, vol. 1 (1955), p. 427).
I shall give here proofs of those of my results which do not overlap with ASSER'S. 1. By a "function" I shall mean always a function from non-negative integers to non-negative integers. The number of arguments of a function is arbitrary but always finite.
Let K be the least class of functions satisfying conditions (1) -(3) given below:
(1) The following functions &, u:, 8, c belong to
( 2 ) If functions fh(xl, . . . , xk, n), fi(yl, . . . , y p , n) are in K , then so is the conipound function (the y i s need not t o be distinct from the x i s )
fh(x1,. * -9 5 -1 7 fi(Y1, 1 --9 yplpt n), x j + l t . * * I xk, n ) .
(3) If functions f h ( z , , . . ., xk, n), f 2 ( z , y, q,. . ., xk, n) are in K and if fi(0, X I , * . * , sk, n) = f h ( x l , . . . > "k, n ) , f i ( Z + l , x l , . . . , x k , n ) = m i n ( f 2 ( z , f i ( z , z 1 , . . . , x k , n ) , z l , . . . , z k , n ) , n ) ,