571 



II, " On Fermat's Theorems of the Polygonal Numbers." First 

 Communication. By The Eight Hon. Sir FREDERICK 

 POLLOCK, F.R.S., Lord Chief Baron. Received July 11, 



1860. 



(Abstract.) 



This paper relates to the second theorem, viz. that which asserts 

 that every number is composed of 4 square numbers (0 [or zero] 

 being considered as an even square). If every odd number be com- 

 posed of 4 square numbers, then every even number must also be 

 composed of 4 square numbers ; for every even number must, on a 

 continued division by 2, ultimately become an odd number. The 

 paper relates chiefly to the Table which accompanies it, from which 

 it appears that a remarkable law obtains as to the division of odd 

 numbers (2n+\) into 4 square numbers when a number of the 

 form 4w -fl is divisible into 2 square numbers, which (as 4n+l is 

 an odd number) must be one of them odd, the other even. Before 

 explaining the Table, it is proper to state that if an odd number be 

 divisible into 4 square numbers, three of them must be odd, and one 

 of them even, or one of them must be odd, and 3 of them even, other- 

 wise their sum cannot be an odd number ; it follows from this that if 

 the difference between any two of them be an odd number, the differ- 

 ence between the other two must be an even number, and vice versa-, 

 for let a 2 -f 6 2 + c 2 + d 2 =2ra+ 1, then if 2 -6 2 =2p, c 2 -^ 2 must equal 

 20-f 1; if possible let c 2 -^=2r, then a 2 6 2 + c 2 -d 2 =2p-f 2r . 

 add 2b 2 + 2d 2 (an even number) to each, and a 2 + 5 2 + c 2 -f- d 2 will be 

 an even number, which by the hypothesis it is not ; if, therefore, 

 # 3 5 2 be an even number, c 2 c? 2 cannot also be an even number, 

 and therefore must be an odd one. If, therefore, the four roots of 

 the squares into which any odd number may be divided are arranged 

 in any order there will be three differences ; the two exterior differences 

 will be one odd, the other even ; the middle difference may be either 

 odd or even. 



The Table is arranged thus : the lowest row of figures is the 

 series 1, 5, 9, 13, 17, &c. (4w+ 1) ; the next row above is the series 

 of natural numbers, 0, 1, 2, 3, 4, &c. (w), &c. ; the next row is 

 1, 3, 5, 7, 9, &c. (2ra+l) the odd numbers; each of the odd 

 numbers is the first term in a series increasing upwards by the num- 



