Vieta to Descartes

(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) $x^{4} + y^{4} = z^{2}$ 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 $a^{p - 1} \equiv 1 (\mod p)$ . 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 $2^{2^{n}} + 1 =$ 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 $2^{2^{5}} + 1 = 4, 294, 967, 297 = 6, 700, 417 times 641$ . 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,

given in the Relation, was used by Fermat for the decomposition of large numbers into prime factors.

(11) If the integers $a$ , $b$ , $c$ represent the sides of a right triangle, then its area cannot be a square number. This was proved by Lagrange.

(12) Fermat's solution of $a x^{2} + 1 = y^{2}$ , where $a$ is integral but not a square, has come down in only the broadest outline, as given in the Relation. He proposed the problem to the Frenchman, Bernhard Frenicle de Bessy, and in 1657 to all

living mathematicians. In England, Wallis and Lord Brounker conjointly found a laborious solution, which was published in 1658, and also in 1668, in an algebraical work brought out by John Pell. Though Pell had no other connection with the

problem, it went by the name of "Pell's problem." The first

solution was given by the Hindoos.

We are not sure that Fermat subjected all his theorems to rigorous proof. His methods of proof were entirely lost until 1879, when a document was found buried among the manuscripts of Huygens in the library of Leyden, entitled Relation des découvertes en la science des nombres. It appears from it that he used an inductive method, called by him la descente infinie ou indefinie. He says that this was particularly applicable in proving the impossibility of certain relations, as, for instance, Theorem 11, given above, but that he succeeded in using the method also in proving affirmative statements. Thus he proved Theorem 3 by showing that if we suppose there be a prime $4 n + 1$ which does not possess this property, then there will be a smaller prime of the form $4 n + 1$ not possessing it; and a third one smaller than the second, not possessing it; and so on. Thus descending indefinitely, he arrives at the number $5$ , which is the smallest prime factor of the form $4 n + 1$ . From the above supposition it would follow that $5$ is not the sum of two squares–-a conclusion