................ ..... g. £. D. VOL. II. PART IL k THIOREM. THEOREM, Every integral number whatever, is either a square, or the sum of two, three, or four squares. This is an immediate consequence of the last proposition and of the formulae at Prop. 3; for every...

...prevent any one OB more of these squares from becoming zero; therefore, every integral number whatever is either a square, or the sum of two, three, or four squares. a. ED Cor. All that has been proved in the foregoing proposition for integral numbers, is equally true...

...(wz' + xy' С — ух'— zwj as will appear by the development of these formulae. 28. Every integral number is either a square, or the sum of two, three, or four squares. The latter is one of the celebrated numerical theorems of Fermât, first demonstrated by Lagrange....

...publication of the Mathematical Questions proposed in the Ladies' Diary. (See vol. if. p. 342.) (38.) Every number is either a square, or the sum of two, three, or four squares. Thus 5=4 + 1, 30=25 + 4 + 1, 63=49+9+4+1, or, 36 + 25 + 1 + 1. This curious property has been demonstrated...

...77»« Mathematical Repoñtory. — Original Papers Continued. Theorem. Every integral number whatever, is either a square or the sum of two, three, or four squares. Proposition I. If A be any prime number, and all the consecutive squares, 1s, 11J, 32, 1-, &c .........

..."This theorem is a special case of a more general one. As to this case and the following one, viz.: Every number is either a square or the sum of two, three, or Jour squares, the demonstrations have been discovered, but as to the pentagonal and higher numbers,...

...to desist from speculation in cases of doubt. (c) Composition of numbers as the sum of four squares. Every number is either a square or the sum of two, three or four squares. This well-known theorem, enunciated by Fermat in his note to Diophantos iv. 31, shows at once that...

...ways. (5) Every number is either a triangular number or the sum of two or three triangular numbers ; either a square or the sum of two, three, or four squares ; either a pentagonal number or the sum of two, three, four, or five pentagonal numbers ; similarly...

...in this case, as he does when it is a question of dividing a number into three or two squares. Now every number is either a square or the sum, of two, three or four squares (a theorem enunciated by Fermat and proved by Lagrange who followed up results obtained by Euler),...

...in this case, as he does when it is a question of dividing a number into three or two squares. ¿ow every number is either a square or the sum of two, three .r four squares (a theorem enunciated by Fermat and proved 1 y Lagrange who followed up results obtained...