150 MEMOIR OF LEGENDRE. 



There arc prime numbers of all mag-nitmles; but when the numbers are some- 

 what o'reat it is not easy to discover immediately whether they are prime or not. 

 The prime numbers arc distributed among' the odd numbers Avith an apparent 

 iiTco-ularitv which is _vet subject to certain laws. The search for them, the deter- 

 mination of the quantities of them which exist in a given interval of the numeric 

 scale, form one of the objects of the theory of numl)ers. 



Numbers may be ranged by scries in each of which may be remarked the con- 

 stant existence of certain properties ; such are the triang'ular numbers 1, 3, G, 10, 

 15, &c., each expressing a number of units which may be arranged triangularly ; 

 the quadratic nun)bers 1, 4, 9, 16, 25, which in the same way correspond to the 

 square; polygonal numbers, pyramidal, &c. ; and these series give rise to com- 

 binations more or less curious. Certain numbers are the squares of other smaller 

 ones, as 4 the square of 2, 9 of 3, &c. ; others, as 8, 13, 18, are the simi of two 

 squares ; others again, like 17 for example, are the sum of three squares, l^agrange 

 and Euler have jiroved that there is iio number ivhkli is not the sum of Jour or 

 of a less number <f squares * 



These properties and many others are at once remarked in examples taken 

 among- numbers of little amount, and it l)ecomes a matter of curiosity to follow 

 them among the larger numbers in order to learn whether they are general or 

 not. Hence proceed researches which are often very difficult and provoke a 

 lively interest. The final conclusion evades detection so much the longer from 

 the circumstance that frequently there exists, as yet, in science ho rule for seek- 

 ing it ; it is a prey which for a long time eludes the pursuit of the hunter. Again, 

 there are certain properties of numbers which come to light unexpectedly in their 

 combinations, and which, presenting something enigmatical and surprising, have 

 been often held to pertain to the mysterious. Hence the virtues which necro- 

 mancers have pretended that they detected in cabalistic numbers ; virtues which 

 are to the theory of numbers not unlike what astrology is to astronomy. 



" It would seem (remarks M. Legendre) that Euler had a peculiar taste for 

 the science of numbers, and that he gave himself up to thib' kind of research 

 with a sort of passion, as happens (he adds) to almost all those who are occupied 

 with it ;" and it is clear that M. Legendre himself formed no exception to this 

 remark. 



The first researches of M. Legendre on numbers, contained in his distinguished 

 memoir of 1785, constituted a direct sequel to those of Euler and Lagrange which 

 they extended and developed in several important particulars ; but JM. Legendre 

 embodied also in this work mau}^ discoveries entirely new, and particularly the 

 theorem of reci2^roc if I/, ]u\owa\\\ie\\\sG under the name of the law of Legendre, 

 one of the most fertile laws of the theory of numbers. 



This theorem, more readily expressed in algebraic than ordinary language,t 



"LegenJre, Tliorie des Nombres, t. I, p. 211. 



t The tollovviu^ are the terms in which M. Legendre enunciates, in the Thcorie dcs Nom- 

 bres, I, 23U, the theorem in question : ij VI. Theorem containing a law of reciprocity ichicli 

 exists between any two prime numbers whatever. (166.^ We liave seen (No. joo) tliat it m 



and « be any two prime numbers, odd and unequal, the abridged expressions ( 7 | ( ,-j 



represent, one the remainder m —^ — the other the remainder n — — divided by in. At 



the same time it has been proved that one and the other remainder can never be other than 

 -f-1 and — 1. This being so, there exists such a relation between the two remainders 



(-1 and I— j that one being known, the other is immediately determined. The following 

 is the general theorem which contains this relation : 



Whatever be the prime numbers in and n, if they are not both of the form 4x-j-'^i ^^♦^ 



shall always have l-j = l- j and if they are both of the form 4x-f-3, we shall have 



(^1 =^ — \-\ These two cases are comprised in the formula 



(n\ / ^ \ m — ' n — ' Im \ 

 m) — \—^} a • -^ • \^.) 



