232 report — 1859. 



numbers required and the numbers given. The arithmetical solution of a 

 problem should consist in prescribing a finite number of purely arithmetical 

 operations (exempt from all tentative processes), by which all the numbers 

 satisfying the conditions of the problem, and those only, are obtained. It is 

 clear that this description exactly applies to the methods on which the 

 solution of linear congruences depends; but, unfortunately, the higher arith- 

 metic presents but few examples of solutions of equal perfection. 



8. Besides the older methods for the solution of the equation ax=\ + ¥i/, 

 others have, in very recent times, been suggested. Of these the following 

 may serve as examples : — 



A. In the equation ax= 1 -f P#> or the congruence a#= 1 , mod P, form the 

 residues of the successive powers of a for the modulus P. If a be prime 

 to P, we shall at last arrive at a power which has + 1 for its remainder or 

 residue. The residue of the power immediately inferior to this power 

 is the value of x in the congruence ax = 1, mod P. This solution is evidently 

 an application of Fermat's Theorem*. 



B. Let there be P points A 1 A 2 . . . A P , arranged at equal distances on the 

 circumference of a circle. Join A a to A a+1 , A a+1 to A 2n+1 ...and so on 

 continually. It can be proved that if a be prime to P, we shall not return 

 again to A 1? until we have passed through every one of the P points, and 

 have formed a polygon of P sides. Let X 1 X 2 ...X? be the vertices of this 

 polygon, taken in order, and let A 2 =X m+1 ; then x^m is the value of x in 



the congruence ax=l, mod Pf. 



C. Let an origin and a pair of axes be assumed in a plane, and let all the 

 points be constructed whose coordinates are integral multiples of the linear 

 unit; call these points unit points. Join the origin to the point (a, P). If 

 a be prime to P, no unit point can lie on the joining line, but on each side 

 of the joining line there will be a point lying nearer to it than any other. 

 Let (£j t) x ), (£ 2 ?? 2 ) be t,ie coordinates of these points, and let ^ : j^ < £ 2 : jj 2 ; 

 then | x , J7 X , and £ a , r\ 2 are the least positive numbers satisfying the equations 

 a j?j— P^=l, ci)i 2 — P£ 2 =— !• 



The late M. Crelle, of Berlin, in the 45th volume of his Journal (p. 299), 

 has given a very useful table, containing the least positive numbers x Y and 

 x 2 which satisfy the equation a x x x —a 2 x 2 —\, for all values of a x up to 120, 

 and for all values of a 2 prime to a r and less than it. 



9. Systems of Linear Congruences. — The theory of these systems is left 

 imperfect in the work of Gauss (see Disq. Arith. art. 37) ; but, by the aid of 

 a few subsidiary propositions relating to determinants, we may, in every case, 

 obtain directly all possible solutions of any proposed system ; and (what is 

 frequently of more importance) we can decide a priori whether a given 

 system of linear congruences be resoluble or not, and if it be resoluble we 

 can assign the number of its solutions. The following theorems by which 

 the determination of the number of solutions is, in every case, effected, will 

 sufficiently indicate the nature of these investigations. 



Let the proposed system of congruences be represented by 



(1, l)x l + (], t 2)x 2 +(l,3)x 3 +..+(l,n)x=u 1 



* Binet, sur la Resolution des equations du premier degre en Nombres entiers. (Journal 

 de l'ficole Polytechnique, cahier xx., p. 289.) 



Libri, Memoires de Mathematique et Physique (Florence, 1829), p. 65-67. 



Poinsot, Reflexions sur les Principes Fondamentals de la Theorie des Nombres (Paris, 

 1845), cap. iii. nos. 19 & 20. For another solution by M. Binet, see Comptes Rendues, 

 xiii., p. 349. See also Cauchy, Comptes Rendues, xii. p. 813. 



t Poinsot, Reflexions, &c, cap. iii., nos. 17 and 18. 



