86 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



A. Chatelet 222 gave a brief summary of results, especially Heger's. 205 

 E. Cahen 223 gave an extended treatment of systems of linear equations, 

 congruences, and linear forms. 



M. d'Ocagne 224 solved x + y + z + t = n, 5x + 2y + z + \t = n to 

 find the number of ways to pay a sum of n francs with 5, 2, 1, ^ franc coins, 

 n in all. For similar problems, see Schubert 143 and d'Ocagne 178 of Ch. III. 



ONE LINEAR CONGRUENCE IN TWO OR MORE UNKNOWNS. 



Th. Schonemann 225 considered the number Q of sets of solutions of 



+ + a m m = (mod p), 



with x, , distinct and with the understanding that the solutions 

 obtained by permuting equal elements a count as a single solution, and 

 p is prime. Let \i of the a's be equal, v further o's be equal, etc. If 

 GI + + a. ^ (mod p] and m ^ p, 



But if i + + a m = (mod p), while the sum of fewer a's is not divisible 

 by p, 



----- 1 -...- 



V ' ' fJLi V 



V. A. Lebesgue, 226 by specialization of his 17 result in Ch. VIII of Vol. I 

 of this History, found that, if p is a primitive root of the prune p, 



P b Xi + p c x z + + p*x k = 0, p a + p b xt + + p*x k = (mod p) 

 each have p k ~ l sets x\, , Xk of solutions ^ 0, but have 



(p - l){(p - I)*"* -(- I)*"!}, {(p - 1)* ~ (- 1)*} 



sets of solutions > 0, respectively. 



M. A. Stern 227 proved that, if p is an odd prime, any integer can be 

 expressed modulo p in exactly P = (2 P ~ 1 l)/p ways as one or the sum 

 of several distinct numbers chosen from the set 1, 2, , p 1. For 

 example, 3 = 1+2 = 1 + 3 + 4 (mod 5). Restricting ourselves to an 

 even number of summands, we find that zero can be expressed in 

 |(P + P ~ 2) ways, while 1, 2, , or p 1 can be expressed in %(P 1) 

 ways. We shall report in the chapter on quadratic residues on his results 

 when the set is 1, 2, , (p l)/2. 



222 Legons sur la th6orie des nombres, 1913, 55-8. 



223 Thdorie des nombres, 1, 1914, 110-85, 204-62, 278, 299-315, 383-7, 405-6. 



224 L'enseignement math., 18, 1916, 45-7. Cf. Amer. Math. Monthly, 26, 1919, 215-8. 



226 Jour, fur Math., 19, 1839, 292. 

 228 Jour, de Math., (2), 4, 1859, 366. 



227 Jour, fur Math., 61, 1863, 66. 



