This does not always happen. In passing, we discussed several special cases from which particular inferences may be drawn. The powerful mathematician Pierre Fermat has presented the world with a theorem of extremely simple form which he discovered, a proof of which is being sought even nowadays, two and a half centuries after he stated it. In easy language, it is this: the sum of two squares may again be a square, for example, 5^2 + 12^2 = 13^2, since 25 + 144 = 169; but the sum of two cubes can _never_ be a cube, and, more generally, as soon as the exponent, the power index _n_, is greater than 2, the equation _x_^_n_ + _y_^_n_ = _z_^_n_ can never be satisfied by whole number values for _x_, _y_, and _z_; it is impossible to find three whole numbers for _x_, _y_, and _z_, which, when substituted in the equation, give a correct result. This is certainly true; it is an intuitive discovery. But Fermat's assertion that he possessed a "wonderful proof," is for very good reasons open to contradiction. No one doubts the absolute truth of the theorem. But the later inspiration, the next step after the intuition, has occurred neither to Fermat nor to anyone else. It cannot be established whether his remark about the proof was due to a subjective error, or was baseless. In any case it seems probable that Fermat had arrived at the result _per intuitionem_ without knowing the way to it. His creative act stopped short; it was only a first flare of a conflagration, and did not fulfil the condition that Einstein associates with the conception of a logically complete method. We may, indeed, pursue this case of Fermat still further. He had enunciated another theorem, again _per intuitionem_, namely, that it was possible to construct prime numbers of any magnitude by a formula he gave. Euler later showed by a definite example that the theorem was false. It was stated in a letter to Pascal written in 1654 in the words: the result of squaring 2 continuously and then adding 1 must in each case be a prime number, that is, 2^(2^_k_) + 1 must always be a prime no matter what value _k_ may have. Fermat added: "This is a property for the truth of which I answer." Euler chanced to try _k_ = 5, and found that 2^32 + 1 = 4,294,967,297, which may be represented as the product of 641 and 6,700,417, and hence is not a prime. It is conceivable that no Euler might have lived, and that no one else might have discovered this contradiction. What would then have been the position of this "discovery" of Fermat?