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 times of Diophantus and the Hindoos until the beginning of the seventeenth century. But the illustrious period we are now considering produced men who rescued this science from the realm of mysticism and superstition, in which it had been so long imprisoned; the properties of numbers began again to be studied scientifically. Not being in possession of the Hindoo indeterminate analysis, many beautiful results of the Brahmins had to be re-discovered by the Europeans. Thus a solution in integers of linear indeterminate equations was re-discovered by the Frenchman Bachet de Méziriac (1581–1638), who was the earliest noteworthy European Diophantist. In 1612 he published Problèmes plaisants et délectables qui se font par lea nombres, and in 1621 a Greek edition of Diophantus with notes. The father of the modern theory of numbers is Fermat. He was so uncommunicative in disposition, that he generally concealed his methods and made known his results only. In some cases later analysts have been greatly puzzled in the attempt of supplying the proofs. Fermat owned a copy of Bachet's Diophantus, in which he entered numerous marginal notes. In 1670 these notes were incorporated in a new edition of Diophantus, brought out by his son. Other theorems on numbers, due to Fermat, were published in his Opera varia (edited by his son) and in Wallis's Commercium epistolicum of 1658. Of the following theorems, the first seven are found in the marginal notes:—

(1) $$\scriptstyle{x^n+y^n=z^n}$$ is impossible for integral values of x, y, and z, when $$\scriptstyle{n>2}$$. Remark: "I have found for this a truly wonderful proof, but the margin is too small to hold it." Repeatedly was this theorem made the prize question of learned societies. It has given rise to investigations of great interest and difficulty on the part of Euler, Lagrange, Dirichlet, and Kummer.

(2) A prime of the form $$\scriptstyle{4n+1}$$ is only once the hypothenuse