260 History of the Theory of Numbers. [Chap, viii 
R. Lipschitz^^^ examined Fermat's^^^ statement and proved that the 
primes p for which a' +1 = (mod p) is impossible are those and only those 
for which a solution u of w" =a (mod p) is a quadratic non-residue of p 
and for which X^A.', where 2^ is the highest power of 2 dividing p — 1. 
Cases when a''+l = is impossible and not embraced by Fermat's rule are 
a = 2, p = 89, 337; a = S, p = 13; a=-2, p = 281; etc. 
L. Kronecker^^^ called /(.r) an invariant of the congruence k=k' (mod m), 
if the latter congruence implies the equality /(/v) =/(//). If also, conversely, 
the equality implies the congruence, f{x) is called a proper (or characteristic) 
invariant, an example being the least positive residue of an integer modulo 
m. It is shown that every invariant of k=k' (mod m) can be represented 
as a symmetric function of all the integers congruent to k modulo m. 
G. Wertheim^^^ proved that a^+l = (mod p) is impossible if a belongs 
to an odd exponent modulo p [Fermat^'^^]. 
E. L. Bunitzky^*^^ (Bunickij) noted that, for any integer M, the con- 
gruences 
f(a+kh)=rk (mod M) (A: = 0, 1,. . ., n) 
hold if and only if the coefficients Ak of /(x) satisfy the conditions 
k\h''Ak=A% (mod M) {k = 1, . . . , n). 
If k is the least value of x for which xlh"" is divisible by M, and if the 
g. c. d. of M and h is k<m, where m is a divisor of M, then if /(a;)=0 (mod 
M) has the roots a, a-\-h,..., a-\-{k — l)h, it has also the roots a-\-jh 
{j = k,k-\-l,...,m-l).^ 
G. Biase^^'' called a similar to h in the ratio m:n modulo k if the remainders 
on dividing a and h by k are in the ratio m:n. Two numbers similar to a 
third in two given ratios modulo k are similar to each other modulo k in 
a ratio equal to the quotient of the given ratios. 
The problem^ '^^ to find n numbers whose n^ — n differences are incon- 
gruent modulo n^ — n+1 is possible for n = 6, but not for n = 7. 
R. D. von Sterneck^^° proved that, if A is not divisible by the odd 
prime p, Ax'^+Bx^+C takes \p{2AB, p) incongruent values (when x ranges 
over the set 0, 1, . . ., p — 1) if 5 is not divisible by p, while if B is divisible 
by p, it takes (p+3)/4 or (p-fl)/2 values according as p = 4n-l-l or p = 
471 — 1. In terms of Legendre's symbol, 
>««Bull. des Sc. Math., (2), 22, I, 1898, 123-8. Extract in Oeuvres de Fermat, 4, 196-7. 
>"Vorlesungen iiber Zahlcntheorie, I, 1901, 131-142. 
"^AnfangsRTunde der Zahlenlehre, 1902, 265. 
''"Zap. mat. otd. Obsc. (Soc. of natur.), Odessa, 20, 1902, III- VIII (in Russian); cf. Fortschr. 
Math., 33, 1902, p. 205. 
•^"Il Boll. Matematica Gior. Sc. Didat., Bologna, 4, 1905, 96. 
i"L'interm6diaire des math., 1906, 141; 1908, 64; 19, 1912, 130-1. Amcr. Math. Monthly, 13, 
1906, 215; 14, 1907, 107-8. 
