On the nature and use of the functions employed in the recognition of quadratic residues
Emory McClintock · Transactions of the American Mathematical Society · 1902
The congruence n_ x2(mod k) is possible, and n is therefore a quadratic residue of k, when n is a quadratic residue of each prime factor of k, so that in order to determilne the possibility of the congruence in all cases we must be able to determine its possibility when k is any prime number. The case k 2 is simple, but when k is an odd prime the problem presents some difficulties, and it has perhaps received more attention than any other in the theory of numbers. LEGENDRE introduced the symbol (n/k) = i 1 E= ny(k-) (mod k), the sign being + or as n is or is not a quadratic residue of the prime number k, and since his time the problem has consisted in determining the sign of (n/k) for any given values of n and k, n being prime to the odd prime k. The method of evaluation, or algorithm, of LEGENDRE, improved by JACOBI, is still the standard solution. It requires the use of the law of quadratic reciprocity formulated by LEGENDRE, though perceived earlier by EULER: theoremafundamentale, as it was called by GAUSS, who first supplied for it a satisfactory demonstration. The derivation of this law has attracted uniusual attention from many mathematicians, eight demonstrations having been prodcuced by GAUSS alone. The chief improvement since the time of JACOBI consists in an observation made independently by SCHERING and KRONECKER4 namely, that 4 GAUSS'S characteristic, ,, is available for the proof of the law of reciprocity when k is not prime. The definition (n/k) (1)', employed by TANNERY in his proof of the usual algorithm, is one of two employed in the present paper, and is herein extended and applied to wider purposes, with only the slightest reference to the law of reciprocity. I find great advantage in substituting for the symbol ,t the broader symbol , (n, k), so as to be able to discuss the function , for different values of n and k, and thereby to develop relations of the fulnctions ,t(n, k) im-