36 ARITHMETICAL RECREATIONS [CH. I] 



Bachet's problem falls under case (ii), n = 40. MacMahon's 

 analysis shows that there are eight such ways of factorizing 

 ar 40 + ar 39 + ... + l + ...+x S9 + x i0 . First, there is the expres- 

 sion itself in which a = 1, m = 40. Second, the expression is 

 equal to (1 — af 1 )/^ (1 — x), which can be resolved into the 

 product of (1 — x*)jx (1 — x) and (1 — a 81 )/^ 39 (1 — X s ) ; hence it 

 can be resolved into two factors of the form given above, in 

 one of which o = l, m = l, and in the other a = 3, m = 13. 

 Third, similarly, it can be resolved into two such factors, in 

 one of which o = l, m = 4, and in the other a =9, m = 4. 

 Fourth, it can be resolved into three such factors, in one of 

 which a = l, m = l, in another a = 3, m = \, and in the other 

 a = 9, m = 4. Fifth, it can be, resolved into two such factors, 

 in one of which a= 1, m = 13, and in the other a = 27, m = l. 

 Sixth, it can be resolved into three such factors, in one of 

 which o=l, m = l, in another a = 3, m = 4, and in the other 

 a = 27, m = l. Seventh, it can be resolved into three such 

 factors, in one of which a = 1, m = 4, in another a = 9, m=l, 

 and in the other a = 27, m= 1. Eighth, it can be resolved into 

 four such factors, in one of which a = 1, m = 1, in another a = 3, 

 m = 1, in another a = 9, m = 1, and in the other a = 27, m = 1. 



These results show that there are eight possible sets of 

 weights with which any integral number of pounds from 1 to 

 40 can be weighed subject to the conditions (ii), (a), and (6). 

 If we denote p weights each equal to w by w p , these eight 

 solutions are l i0 ; 1, 3 13 ; l 4 , 9 4 ; 1, 3, 9 1 ; l 18 , 27; 1, S\ 27; 

 l 4 , 9, 27; 1, 3, 9, 27. The last of these is Bachet's solution: 

 not only is it that in which the least number of weights are 

 employed, but it is also the only one in which all the weights 

 are unequal. 



Problems in Higher Arithmetic Many mathematicians 

 take a special interest in the theorems of higher arithmetic: 

 such, for example, as that every prime of the form 4w + 1 and 

 every power of it is expressible as the sum of two squares*, and 

 that the first and second powers can be expressed thus in only 

 one way. For instance, 13 = 3 2 + 2 2 , 13 2 = 12 2 + 5 2 , 13 s = 46 2 + 9 J , 



* Fermat's Diophantus, Toulouse, 1670, bk. in, prop. 22, p. 127; or 

 Brassinne's Precis, Paris, 1853, p. 65. 



