CHAP. II] ONE LINEAR CONGRUENCE IN Two OR MORE UNKNOWNS. 87 



E. Lucas 228 noted that, if a is prime to n, the points (x, y}, where x = 0, 

 1, , n and y is the residue modulo n of ax, lie on a lattice (composed of 

 equal parallelograms), and are said to form a satin n a . These satins lead 

 graphically to all solutions of mx + ny = (mod p). 



L. Gegenbauer 229 gave a direct proof of Lebesgue's 226 results. Let the 

 number of sets of solutions each ^ of a&i + + a k x k + 6 = (mod 

 p), where each a is not divisible by the prime p, be S' k or S k according as 

 6 is or is not divisible by p. Let N be the number of all sets of solutions. 

 Since a k x k + 6 ranges with x k over a complete set of residues modulo p, 

 N is the sum of the numbers of sets of solutions of the p congruences 

 diXi + + a k -iX k --L + c = (mod p}, c = 0, 1, , p 1; while the 

 number of those of these sets of solutions whose elements are prime to p 

 equals the sum of the numbers of the sets of solutions of like property of the 

 p 1 congruences aiXi + + ak-iX k -i + c' = 0, c' = 0, , 6 1, 

 6 + 1, , p 1. Hence 



K. Zsigmondy 230 proved that, according as a. is not or is divisible by the 

 prime p, k + k\ + + & p _i = a (mod p) has \f/(p 1) or $(p 1) 1 

 sets of solutions in which each ki is prime to p, where $(ri) is the number of 

 congruences of degree n with no integral root modulo p. The system of 

 congruences 



&()+ + k p -i = 0, ki + 2k>2 + + (p l)k p -i = a (mod p) 



has \l/(p 2) or $(p 2) + p 1 sets of solutions prime to p according 

 as a. ^ or a. = 0. 



R. D. von Sterneck 231 found the number (ri)t of additive compositions 

 of n modulo M formed of i summands which are incongruent modulo M, 

 i. e., the number of solutions of 



n = xi + x 2 + + xt (mod M), ^ x\ < x 2 < - - < x { < M. 



Let (n)f denote the corresponding number when each summand is not 

 divisible by M, so that < Xi < < a;,- < M. Define f(n, d) to be zero 

 if any prime occurs in d with an exponent which exceeds by at least 2 its 

 exponent in n; but when the primes pi, , PJ occur in d with exponents 

 which exceed by unity their exponents in n, and the remaining prime 

 factors of d occur in n at least to the same power as in d, let 



( lV<f)(d) 



f(n, d) = r- . 

 (Pi I)--- (PJ 1) 



228 Application de 1'arith. a la construction de 1'armure des satins r6guliers, Paris, 1868. 



Principii fondamentali della geometria dei tessuti, 1'Ingegnere Civile, Turin, 1880; French 

 transl. in Assoc. frang. av. sc., 40, 1911, 72-87. See S. Giinther, Zeitschr. Math. Naturw. 

 Unterricht, 13, 1882, 93-110; A. Aubry, I'enseignement math., 13, 1911, 187-203; Lucas 106 

 of Ch. VI. 



229 Sitzungsber. Akad. Wiss. Wien (Math.), 99, Ila, 1890, 793^. 



230 Monatshefte Math. Phys., 8, 1897, 40-1. 



231 Sitzungsber. Akad. Wiss. Wien (Math.), Ill, Ila, 1902, 1567-1601. By simpler methods, 



and removal of the restriction on the modulus M, ibid., 113, Ila, 1904, 326-340. 



