150 MEMOIR OF LEGENDRE. 



There are prime numbers of all magnitudes ; but when the numbers are some- 

 what great it is not easy to discover immediately whether they are prime or not. 

 The prime numl»ers are distributed among the odd numbers with an apparent 

 irregularity which is yet 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 numbers. 



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

 stant existence of certain properties ; such are the triangular numbers 1, 3, 6, 10, 

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

 the quadratic numbers 1, 4, 9, 16, 25, which in the same way conx'spond to the 

 squai'e ; 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 sum of two 

 squares ; others again, like 17 for example, are the sum of three squares. Lagrange 

 and Euler have proved that there is no number u-JiicJi is not the sum of four or 

 of a less number of squares* 



These properties and man}- others are at once remarked in examples taken 

 among numbers of little amount, and it becomes a matter of curiosity to follow 

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

 not. Hence proceed researches ^\•hich 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 no 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 numV)crs 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 this kind of research 

 with a sort of ])assion, 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 M. Legendre 

 embodied also in this work many discoveries entirely new, and particularly the 

 theorem of reeiprocitii, known likewise under the name of the law of Lefjendre, 

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



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



"Legendre, Theorie des Numbres, t. I, p. 211. 



t The following are the terms in which M. Legendre enunciates, in the Tlieorie dcs Nom- 

 bres, I, 2:j0, the theorem in question : "J VI. Theorem containing n law of reciprocity icJiich 

 exists between any tico prime numbers wfiatever. (IGG.) We have seeft (No. 135) that it in 



and n be any two prime numbers, odd and unequal, the abridged expressions ( y ) \m} 



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



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



is the general theorem which contains this relation : 



Whatever be the prime numbers m and «, if they are not both of the form, 4 x-|-3, we 



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



I - j = — I- j These two cases are comprised in the formula 



(n\ / \ m — ' 11 — ' hn \ 



m) — \—^) » • ^ • \n.) 



