236 report— 1859. 



Euler*, who was the first to demonstrate Fermat's Theorem, and to develope 

 the numerous arithmetical trutli3 connected with it. 



I. If e and f be conjugate divisors of p — 1 so thatj9— l=ef; the con- 

 gruence #/= 1, mod p, always admits of f incongruous roots. Let these 

 roots be denoted by a x a 2 . ..a/. Then each of the f congruences x e ^a r 

 admits of e solutions, and the ef roots of these/ congruences exhaust com- 

 pletely the p — 1 residues prime to p. It appears, therefore, that if we raise 

 the residues of p to the power e, they will divide themselves into /groups of e 

 numbers apiece ; the e numbers of each group giving, when raised to the 

 power e, the same residue for the modulus p. The numbers a x a 2 . . . a/, are 

 termed the quadratic, cubic, biquadratic, quintic, &c, residues of p, accord- 

 ing as e=2, e=3, e=4, e=5, &c, because they are each of them congruous 

 to an e th power (and indeed to an e th power of e different numbers), and 

 because no other number beside them can be congruous to such a power. 

 Thus every uneven prime has|(/>— 1) quadratic, and as many non-quadratic 

 residues; every prime of the form 4w+l has ^(p—l) biquadratic residues, 

 and three times as many non-biquadratic residues, &c. 



II. It is readily seen that if the same number x satisfy the two congruences 

 xh = 1, and xfr = 1, it also satisfies the congruence x d = 1, modp ; where d 

 is the greatest common divisor of / and/. If therefore / be the lowest 

 index for which the number x satisfies the congruence ;*/= 1, mod p, /is 

 a divisor of p— 1 ; as indeed appears directly from Euler's second demon- 

 stration of Fermat's Theorem. Let \p (f) denote the number of num- 

 bers less than f and prime to it ; then there are always \p (/) roots of the 

 congruence #/= 1, modjp, which cannot satisfy any other congruence of 

 lower index, and similar form. These are called primitive roots of the con- 

 gruence #/ = 1, mod p ; they are also said to appertain to the exponent/. If 



/=/>— 1> the \p(p — 1) primitive roots of the congruence xP~ l = 1, modp, are 

 termed for brevity (though the designation is somewhat improper) the pri- 

 mitive roots of p. There are therefore \p (p — 1) primitive roots of p. 



13. Primitive Roots. — The problem of the direct determination of the pri- 

 mitive roots of a prime number is one of the " cruces "of the Theory of Num- 

 bers. Euler, who first observed the peculiarity of these numbers, has yet left 

 us no rigorous proof of their existencef ; though assuming their existence he 

 succeeded in accurately determining their number. The defect in his de- 

 monstration was first supplied by Gauss {, who has also proposed an indirect 

 method for finding a primitive root. This method § consists in taking any 

 residue a of p, and determining (by the successive formation of its powers) 

 the exponent/ to which it appertains. If/=p — 1, a is itself a primitive 

 root of p ; if not, let b be a second residue of p, not contained in the period 

 of a, (i. e. not congruous for the modulus p to any one of the numbers a , 

 a, a 2 , . . . .a/- 1 ,) and let the exponent to which b appertains be determined. 

 This exponent cannot (as is shown by Gauss) be identical with, nor yet a 



* Euler's memoirs on this Theory are, — 



(i). Theorematum quorundam ad numeros primos spectantium demonstratio. Comment. 

 Arith. vol. i. p. 21. 



(ii). Theoremata circa residua ex divisione potestatum relicta. Ibid. p. 260. 



(iii). Theoremata arithmetica novo methodo demonstrata. Ibid. p. 274. 



(iv). Disquisitio accuratior circa residua ex divisione quadratorum aliarumque potestatum 

 per numeros primos relicta. Ibid. p. 487. 



(v). Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia. 

 Ibid. p. 516. 



t See the memoir (i) of the preceding note ; and Gauss's criticism on it ; Disq. Arith. 

 Art. 56. 



J Disq. Arith. Art. 52-55. § Ibid. Art. 73-74. 



