"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square."

Pierre de Fermat Mathematics See all quotes by Pierre de Fermat →

React to this quote

More by Pierre de Fermat

“The result of my work has been the most extraordinary, the most unforeseen, and the happiest, that ever was; for, after having performed all the equations, multiplications, antitheses, and other operations of my method, and having finally finished the problem, I have found that my principle gives exactly and precisely the same proportion for thes which Monsieur Descartes has established.”

Resonant Thoughts

"The vulgar opinion, then, which, on health reasons, condemns vegetable food and so much praises animal food, being so ill-founded, I have always thought it well to oppose myself to it, moved both by e…"

— Antonio Cocchi

Science1695–1758

#knowledge

"These two states which it is necessary to know together in order to see the whole truth, being known separately, lead necessarily to one of these two vices, pride or indolence, in which all men are in…"

— Blaise Pascal

Mathematics1623–1662

#truth

"As you probably know, I'm often accused of intellectual exhibitionism and all forms of elitism. Although I can understand this point of view, it's a rather wasted argument because, if we regard areas …"

— Peter Greenaway

Art1942

#knowledge

"I am aware that the following verdict is extremely unpopular among both politicians and intellectuals, but they should learn to face facts at last. The division of guilt for India's Hindu-Muslim confl…"

— Koenraad Elst

Indology1959

#knowledge