Roots of Polynomials Modulo Prime Powers

To say that R is a root set modulo n means that R is a subset of Z n , the ring of integers modulo n , and there is a polynomial the roots of which modulo n are exactly the elements of R . Note that [ and Z n are always root sets modulo n . It seems that only two papers have appeared which mention the nature of root sets modulo n , and then only at a very basic level : Sierpin ́ ski [3] and Chojnacka-Pniewska [1] noted that not every subset of Z 6 is a root set modulo 6 . Of course , for a prime p , every subset of Z p is a root set modulo p , but , in general , it appears that the property of being a root set modulo n is rare . The theorems of the next section provide tools that permit the ef ficient computation of the number of root sets modulo a prime power . Throughout this note , p is a prime and k is a positive integer . For an integer j and an integer m > 1 , j m , read j to the m falling , is defined by j m 5 j ( j 2 1)( j 2 2) ? ? ? ( j 2 m 1 1) . Also j 0 I is defined to be 1 . For an integer n > 1 , and a prime p , » p ( n ) will denote the highest power of p that divides n . It is well known (see Graham , Knuth and Patashnik [2] , for example) , that