260 HlSTOEY OF THE THEOBY OP NUMBERS. [CHAP. VIII
R. Lipschitz166 examined Fermat's148 statement and proved that the primes p for which a*+l=0 (mod p) is .impossible are those and only those for which a solution u of w2*=a (mod p) is a quadratic non-residue of p and for which X^fc, where 2* is the highest power of 2 dividing p — 1. Cases when az4-ls=0 is impossible and not embraced by Fermat's rule are o=2, p = 89, 337; a = 3, p = 13; a= -2, p = 281; etc.
L. Kronecker167 called /(x) an invariant of the congruence fc==fc' (mod m), if the latter congruence implies the equality /(fc) =/(&'). If also, conversely, the equality implies the congruence, /(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 &==&' (mod m) can be represented as a symmetric function of all the integers congruent to k modulo m.
G. Wertheim168 proved that a^+l^O (mod p) is impossible if a belongs to an odd exponent modulo p [Fermat148].
E. L. Bunitzky169 (Bunickij) noted that, for any integer M, the congruences
f(a+kK)=rk (mod M) (k = 0, 1, . . . , ri)
hold if and only if the coefficients Ak of f(<c) satisfy the conditions k ! fc*Afc3 A*TO (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 fc<w, where w is a divisor of M, then if /(s)==0 (mod M) has the roots a, a+/i, ..., a+(k—l)h, it has also the roots a+jh tf-fc, *+!,... ,m-l).
G. Biase170 called a similar to 5 in the ratio m:n modulo k if the remainders on dividing a and b by fc 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 problem171 to find n numbers whose n2—n differences are incon-gruent modulo n2— n+1 is possible for n = 6, but not for n — 1.
R. D. von Sterneck140 proved that, if A is not divisible by the odd prime p, Ax*+Bx2+C takes \f/(2AB, p) incongruent values (when x ranges over the set 0, 1, . . ., p—1) if B is not divisible by p} while if B is divisible by p, it takes (p-h3)/4 or (p-\-l)/2 values according as p=4nH-l or p = 4n—l. In terms of Legendre's symbol,
186Bull. dee Sc. Math., (2), 22, 1, 1898, 123-8. Extract in Oeuvres de Fermat, 4, 196-7.
167Vorlesungen iiber Zahlentheorie, I, 1901, 131-142.
188Anfangsgriinde der ZahlenJebre, 1902, 265.
189Zap. mat. otd. Obsc. (Soc. of natur.), Odessa, 20, 1902, III- VIII (in Russian); cf. Fortschr.
Math., 33, 1902, p. 205.
170I1 Boll. Matematica Gior. Sc. Didat., Bologna, 4, 1905, 96. ^L'interme'diaire des math., 1906, 141; 1908, 64; 19, 1912, 130-1. Amer. Math. Monthly, 13,
1906, 215; 14, 1907, 107-8.