1. On the extension of the principle of Fermat’s theorem of the polygonal numbers to the higher orders of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders

Abstract

The object of this paper professes to be to ascertain whether the principle of Fermat’s theorem of the polygonal numbers may not be extended to all orders of series whose ultimate differences are con­stant. The polygonal numbers are all of the quadratic form, and they have (according to Fermat’s theorem) this property, that every number is the sum of not exceeding, 3 terms of the triangular num­bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. It is stated in this paper that the series of the odd squares 1,9,25,49, &c. has a similar property, and that every number is the sum of not exceeding 10 odd squares. It is also stated, that a series con­sisting of the 1st and every succeeding 3rd term of the triangular series, viz. 1,10,28,35, &c., has a similar property; and that every number is the sum of not exceeding 11 terms of this last series, and that this may be easily proved [it was proved in a former paper by the same author]. The term “Notation-limit” is applied to the num­ber which denotes the largest number of terms of a series necessary to express any number; and the writer states that 5,7,9,13,21 are respectively the notation-limits of the tetrahedral numbers, the octa­hedral, the cubical, the eicosahedral and the dodecahedral numbers; that 19 is the notation-limit of the series of the 4th powers; that 11 is the notation-limit of the series of the triangular numbers squared, viz. 1,9,36,100, &c., and 31 the notation-limit of the series 1,28,153, &c. (the sum of the odd cubes), whose general expression is 2 n 4 — n 2 .