853 



equal 8, or any multiple of 8 (A, B, C, D, <&c., z, y, x, &c , may be 

 any positive whole numbers). 



9. It is a corollary from this, that in any system of notation having 

 an odd square for its base, the sum of the digits will have the same 

 number of odd squares as the number itself; the number of odd 

 squares being in each case the minimum. 



10. Any number of the form 87^ + 4 is composed of 4 odd squares. 



11. It follows from this that every number is composed of 4 tri- 

 angular numbers, or 3, 2 or 1. 



12. From this it is shown that every number is of the form 



13. And that every number of the form 4?? + 2 (if n be greater 

 than 2) is composed of 4 square numbers, 2 even and 2 odd. 



14. And that every number greater than 27 is composed of 8 

 squares exactly. 



15. Every number (beyond a certain small limit) is composed of 



2 triangular numbers + a square number, or 2 triangular numbers 

 + a double triangular number. 



16. A proof is then offered that in the equation 8w + 4=4 odd 

 squares, one of the four odd squares may be any odd square less than 

 8/2 + 4, and therefore 1 may be one of the 4 odd squares ; and if so, 



87i + 4 = l + 3 odd squares, 



.'.8/2 + 3 = 3 odd squares, 

 from which Fermat's theorem of the triangular numbers is an im- 

 mediate corollary. 



17. A proof (by a tabular series) is then suggested, that all the 

 other cases of Fermat's theorem may be deduced from the first, and 

 that it is not necessary to use more than four terms greater than 

 unity (as discovered by M. Cauchy, see Suppl. to Legendre's 

 Theorie des Nombres, p. 21, 22). 



18. A general expression for a succession of series is then given — 



l,(p + l), (Sp + l), (6p+l} &c. (^'l:ill'V + l); 



and it is proved that any number may be composed of not exceed- 

 ing/? +2 terms of the series, of which three only are required to be 

 greater than imity. If p = 9, the series is 1, 10, 28, 55, &c., that is 

 every third triangular number beginning with 1 ; and every number 

 of the form 9n + q consists of ^triangular numbers not divisible by 



3 (if q be greater than 2). 



If jr?=8, the series is 1, 9, 25, 49, &c. 



19. (The odd squares) and every number may he composed, of not 

 exceeding 10 odd squares, \{'p = 6, the series becomes 1, 7, 19, 37 &c. 

 (the difi'erences between the cubes). From the continued addition 

 of the terms of this series, the cube numbers may be formed. 



If ^=4, the series is 1, 5, 13, 25 &c., or 



1,(1+4), (4 + 9), (9 + 16), &c. 



The continued addition of the terms of this series forms the octo- 

 hedral numbers, viz. 1, 6, 19, 44, &:c. 



4* 



