Page:Proceedings of the Royal Society of London Vol 7.djvu/19



The following communications were read :

I. A paper entitled, "Continuation of the subject of a paper read Dec. 22, 1853, the supplement to which was read Jan. 12, 1854, by Sir FREDERICK POLLOCK, &c. ; with a proof of Fermat's first and second Theorems of the Polygonal Numbers, viz. that every odd number is composed of four square numbers or less, and of three triangular numbers or less." By Sir FREDERICK POLLOCK, M.A., F.R.S. &c. Received February 23, 1854.

The object of this paper is in the first instance to prove the truth of a theorem stated in the supplement to a former paper, viz. " that every odd number can be divided into four squares (zero being considered an even square) the algebraic sum of whose roots (in some form or other) will equal 1, 3, 5, 7, &c. up to the greatest possible sum of the roots." The paper also contains a proof, that if every odd number '2n+ can be divided into four square numbers, the algebraic sum of whose roots is equal to 1, then any number n is composed of not exceeding three triangular numbers.

The general statement of the method of proof may be made thus : two theorems are introduced which connect every odd number with the gradation series, 1, 3, 7, 13, &c., of which the general term is n _l_ w 2_f_i or 4p 2 + 2p + l (that is, the double of a triangular number + 1), each term of which series can be resolved into four squares, the algebraic sum of the roots of which, p,p,p,p-{-, or p l,p,p,p may manifestly be = 1. By these theorems it is shown that every odd number is divisible into four squares, having roots capable of