170 REPORT — 1860. 



is completely determined. (See Dirichlet's memoir on the Arithmetical Pro- 

 gression, sect, 7, in the Berlin Memoirs for 1837.) 



78. Primitive Roots of the Powers of Complex Primes. — Diriclilet has 

 shown* that, in the theory of complex numbers of the form a + bi, the powers 

 of primes of the second species (see art. 25) have primitive roots ; in fact, if 

 a + bi be such a prime, and N (a + bi) — a~ + b' 2 =p, every primitive root of 

 p m is a primitive root of (a + bi) m . On the other hand, if q be a real prime 

 of the form 4n-|-3, q m has no primitive loots in the complex theory. Fur in 

 general, if M be any complex modulus, and M=a a i' 3 c"/ . . ., a, b, c, . . being 

 different complex primes, and if A = N(a), B = N (b), C = N(c), etc, the 

 number of terms in a system of residues prime to M, is A a_l (A — 1) B^ _1 



(B— 1) C y_1 (C— 1) And if we denote this number by \L (M), every 



residue prime to M will satisfy the congruence 



#^< M > = 1, mod M, 

 which here corresponds to Eider's extension of Fermat's theorem. If M= q m , 



this congruence becomes xt m ~ (v -0 = 1, mod q m ; but it is easily shown 



that every residue prime to q m satisfies the congruence xQ ^ q — D^], 

 mod q m ; i. e., q m has no primitive roots, because the exponent q m ~ l (q 2 — 1) 

 is a divisor of, and less than, q^( m - 1 )(q 2 — ]). Nevertheless two numbers y 

 and y', can always be assigned, of which one appertains to the exponent q m ~ l 

 (q 2 — 1) and the other to the exponent q m ~ 1 , and which are such that no 

 power of either of them can become congruous to a power of the other, 

 mod q m , without becoming congruous to unity; from which it will appear 

 that every residue prime to q m may be represented by the formula y" y'v, if we 

 give to x all values from to (q 2 — 1) q m - 1 — 1 inclusive, and toy all values 

 from to q m ~ l — 1 inclusive. 



The corresponding investigations for other complex numbers besides those 

 of the form a + bi have not been given. 



We here conclude our account of the Theory of Congruences. The 

 further continuation of this Report will be occupied with the Theories of 

 Quadratic and other Homogeneous Forms. 



Additions to Part I. 



Art. 16. Legendre's investigation of the law of reciprocity (as presented in 

 the ' Theorie des Nombres,' vol. i. p. 230, or in the ' Essai,' ed. 2, p. 198) is 

 invalid only because it assumes, without a satisfactory proof, that if a be 

 a given prime of the form 4ra + l, a prime b of the form 4w + 3 can always 



be assigned, satisfying the equation j - )= — 1 . M. Kummer (in the Memoirs 



of the Academy of Berlin for 1859, pp. 19, 20) says that this postulate is 

 easily deducible from the theorem demonstrated by Dirichlet, that every 

 arithmetical progression, the terms of which have no common divisor, con- 

 tains prime numbers. It would follow from this, that the demonstration of 

 Legendre (which depends on a very elegant criterion for the resolubility or 

 irresolubility of equations of the form ax 2 + by 2 -\-cz 2 = 0) must be regarded 

 as rigorously exact (see, however, the " Additamenta" to arts. 151, 296, 297 

 of the Disq. Arith.). In the introduction to the memoir to which we have 

 just referred, the reader will find some valuable observations by M. Kummer 

 on the principal investigations relating to laws of reciprocity. 



* See sect. 2 of the memoir, Untersuchungen iiber die Thecrie der complexen Zablen, in 

 the Berlin Memoirs for 1841. 



