Page:A History of Mathematics (1893).djvu/199

 of a right triangle; its square is twice; its cube is three times, etc. Example: $$\scriptstyle{5^2=3^2+4^2}$$; $$\scriptstyle{25^2=15^2+20^2=7^2+24^2}$$; $$\scriptstyle{125^2=75^2+100^2=35^2+120^2=44^2+117^2}$$.

(3) A prime of the form $$\scriptstyle{4n+1}$$ can be expressed once, and only once, as the sum of two squares. Proved by Euler.

(4) A number composed of two cubes can be resolved into two other cubes in an infinite multiplicity of 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 for polygonal numbers in general. The proof of this and other theorems is promised by Fermat in a future work which never appeared. This theorem is also given, with others, in a letter of 1637(?) addressed to Pater Mersenne.

(6) As many numbers as you please may be found, such that the square of each remains a square on the addition to or subtraction from it of the sum of all the numbers.

(7) $$\scriptstyle{x^4+y^4=z^4}$$ is impossible.

(8) In a letter of 1640 he gives the celebrated theorem generally known as "Fermat's theorem," which we state in Gauss's notation: If p is prime, and a is prime to p, then $$\scriptstyle{a^{p-1} \equiv 1 \!\!\!\! \pmod{p}}$$. It was proved by Euler.

(9) Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula $$\scriptstyle{2^{2^n}+1}$$ = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example $$\scriptstyle{2^{2^5}+1}$$ = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors, but was unable to explain the method by which he made his marvellous mental computation.

(10) An odd prime number can be expressed as the difference of two squares in one, and only one, way. This theorem,