







J. B. Luce on the Theory of Numbers. 57 



and assuming /9 = « — 3, it is # = 3 (2a — 3), y = 2a(«-3), 

 which gives x = 15, 21, 27, 33, 39, &c. 



and 2/ = 8, 20, 36, 5G, 80, &c. 



D 



fi=a — 5. we find 



# = 35, 45, 55, 65, &c- 



y = 12, 28, 48, 72, &c, and so of others. 



This problem, of course, has been most satisfactorily answer- 

 ed by all who have written on the theory of numbers ; and I 

 have proposed it only as an illustration of our theorem. But as 

 a further illustration, let us come to some new problems. 



Let it be required to find three numbers such, that the square 

 of the first shall bear the same arithmetical proportion to the 

 square of the second, as this does to the cube of the third. 



Here we have to solve the equation x 2 +z 2 =2y 2 , or x 2 - 2y 2 



, T a 2 (52 n _„2\ a 



-z 3 . 1 assume n—^-\ -p , which gives a=-3, and m 



5o a 2 - I put these values of a, and m, in our third converging 



fraction, and find z = ct 3 -\-3(*P 2 n y and y=3« 2 (?+<? 3 ft, where «, 0, 

 n, may be any number ad libitum. But in the present example 

 ft~2, and in order to have the least integral numbers that satisfy 

 the conditions of the problem, we must take « — 1, £=2; and 

 the result is *aft5, y = 22, and z = 7. 



It is required to find two arithmetical proportions, each com- 

 posed of the four least possible numbers, such that the sum of 

 the terms of the one shall be equal to the sum of the terms of 

 the other; and such at the same time, that the sum of the cubes 

 of the two middle terms subtracted from the sum of the cubes 

 of the two extremes, shall, in each case, leave the same cube. 



Answer, 19, 17 : 7, 5. 



17, 13 : 11, 7. 



I do not work these problems, my object being to come to the 

 equation p*— nq*s=l } which is of greater importance. I only 

 mention them to show the nature of problems that this method 

 solves with the greatest ease : and indeed there is scarce any in- 

 determinate problem of the second, and higher degrees, that it 



^ould not solve : 



haps 



become, in better hands, the fundamental one of the theory of 

 number?, especially if it solves the equation/? 2 -wy a -l. 



Since n = a 2 ±b, and m=~r, it follows evidently that if -j is 

 a whole number, by placing its value in the second converging 



s . fl/ttrfcl 



iraction — ^— , we shall obtain p* - nq s « L For instance, let 



2x3 



w= 11 =9+ 2, from which we infer a=3, m—~^-^^. Hence 



Second Series, Vol. VIII, No. 22.— July, 1 849. 8 





