230 History of the Theory of Numbers. [Chap, viii 
K. Zsigmondy^^ proved that, if p is a prime, there are exactly 
,K«, k) = p- - Q p-' + g) p'-' -...+(- 1)» g) 
congruences x"+ . . . = (mod p) not having as roots k given distinct num- 
bers. Also, 
^(n, k)=p4^(n-l, k) + {-ir(^^, rPin, k + l)=^|^{n, A:)-^(n-l, k). 
If n'^k, \J/{n, ^)=p"~''(p — 1)^'. For n = k, \p{n, k) is the number \J/{n) of 
congruences of degree n with no root. The number with exactly i roots is 
{'])\l/{n — i). There are {^V)\l/ii—r) distinct matrices (3) of rank i such 
that Qr-i is the first one of Qq, ai,. . . not divisible by p. 
K. Zsigmondy^^ considered a function $(/) of a polynomial f{x) such 
that $ is unaltered when the coefficients of f{x) are increased by integral 
multiples of the prime p. Let/t^'^(x), i = l,. . ., p'', denote the polynomials 
of degree k which are distinct modulo p and have unity as the coefficient 
of x''. It is stated that 
p" p- 
r,n-2 
i, i' J = 1 
where a takes those values 1, 2, . . ., p" for which /°^(x)=0 (mod p) does 
not have as a root one of the given incongruent numbers ai, . . ., a/, while, 
in the outer sums on the right, i, i',. . . range over the combinations of 
1, . . ., s without repetitions. 
Zsigmondy^^ had earlier given the preceding formula for the case in 
which tti,. . ., a, denote 0, 1,. . ., p — \. Then taking <I>(/) = 1, we get the 
number of congruences of degree n with no root (Zsigmondy^^) . Taking 
$(/) =/, we see that the sum of the congruences of degree n with no root is 
= (mod p), aside from specified exceptions. Taking $(/)=co-^, where co 
is a pth root of unity, and n^p, we see that the system /j;^(x) takes each 
of the values 1, . . . , p — 1 (mod p) equally often. 
Zsigmondy^^ proved his^^'^^ earher formulas, obtained for an integral 
value of X the number of complete sets of residues modulo p into which 
fall the values of the fH (^) not having prescribed roots, and investigated 
the system 5„ of the least positive residues modulo p of the left members 
of all congruences of degree n having no root. In particular, he found how 
often the system B^ contains each residue, or non-residue, of a gth power. 
He investigated (pp. 19-36) the number of polynomials in x which take k 
prescribed residues modulo p for k given values of x. 
3«Sitzung8ber. Ak. Wiss. Wien (Math.), 103, Ila, 1894, 135-144. 
»'Monatshefte Math. Phys., 7, 1896, 192-3. 
"Jahresbericht d. Deutschen Math. Verein., 4, 1894-5, 109-111. 
"Monatshefte Math. Phys., 8, 1897, 1-42. 
