ON THE THEORY OF NUMBERS. 237 



divisor of, the exponent to which a appertains ; but it is always possible by 

 a comparison of the values of a and b to determine a third number, c, which 

 shall appertain to an exponent divisible by each of the exponents to which a 

 and b appertain. By proceeding in this way we shall evidently obtain num- 

 bers appertaining to exponents continually higher, till at last we come to a 

 number appertaining to the exponent jo — 1 ; i. e. to a primitive root of jo. 



M. Poinsot* proposes the following method. If 2, q x ,q^.... &c. be all 

 the prime divisors ofjo— 1, raise the numbers + ], +2, +3,... + ^(jo — 1), 

 which form a system of residues prime to p, to the powers of which the in- 

 dices are 2, q v q.,, &c; so as to determine all the quadratic residues of p, and 

 its residues of the powers q lt q 2 , &c. If from the system of residues 1, 2, 3, 

 • • 'P— 1 » we successively exclude these residues of squares and higher powers, 

 we shall have \p(p— 1) numbers left, which cannot be congruous to any 

 power having an index that divides jo—1, and which are consequently (as may 

 easily be shown) the primitive roots of p. 



This method is very symmetrical ; and if the problem proposed be to find 

 all the primitive roots of p, it is sufficiently direct. But it is (like many- 

 other direct methods in the Theory of Numbers) of interminable prolixity; 

 and becomes absolutely impracticable if p be a number even of moderate 

 size, as it requires us to form the residues of the successive powers of the 

 numbers 1, 2, 3 .. .1 (p— 1). Of course, in performing this operation, the 

 multiples of p are to be rejected as fast as they arise; but, notwithstanding 

 this abbreviation, and others which a little experience will readily suggest, 

 Gauss's method is, for any practical purpose, greatly preferable. 



In a memoir by M. Oltramare in Crelle's Journal (vol. xlix. p. 161), several 

 considerations are offered for facilitating the determination of the primi- 

 tive roots of primes in numerous special cases. Some, however, of the 

 general results of this memoir are erroneous, at least in expression, and the 

 demonstrations of the more particular conclusions contained in it involve no 

 new principle, but may be obtained by combining the definition of primitive 

 roots with the criteria by which (as we shall hereafter see) we are enabled 

 to decide on the quadratic or cubic characters of the residues of given 

 primes. The following may serve as examples of the very interesting results 

 which are thus obtained by M. Oltramare. 



"If a be a prime number and c 2a + l be also a prime, 2 or a is a primitive 

 root of 2a -fl, according as a is of the form 4w + l or 4ra + 3." Thus 2 is 

 a primitive root of 37 and of 83, 11 is a primitive root of 23, 83 of 167, &c. 



" If a be a prime number, other than 3, and if p=2 m o + l, where m is 

 >- 1, be also a prime, 3 is a primitive root of p, unless the congruence 3 2m_1 

 + 1 =0, mod jo, be satisfied." Thus 3 is a primitive root of 89, and of 137. 



Theorems of the same character will be found in the Theorie des Nom- 

 brest of M. Desmarest. By their aid M. Desmarest has constructed a table 

 giving a primitive root for every prime less than 10,000. 



l<t. Indices. — If y be a primitive root of p, the least positive residues of 

 the jo — 1 successive powers of y, 



y\ y 2 > y\ y p ~ l 



which we may denote by 



y v y 2 , y 3 , ... y P -i, I, 



arc all incongruous for the modulus jo. These residues, therefore, irrespective 

 of the order in which they occur, coincide with the numbers 1, 2, 3 . . . jo— 1, 



* Reflexions sur la Theorie des Nombres, cap. iv. art. 3. 

 t Paris, 1852. See pp. 275-279. 



