Prime number

", eleven years younger than Archimedes, was a native of Cyrene. He... measured the obliquity of the ecliptic and invented a device for finding prime numbers. ...In his old age he lost his eyesight, and on that account is said to have committed suicide by voluntary starvation."

"The primes are the atoms of arithmetic, the indivisible building blocks from which all other numbers are constructed. In their chaotic distribution lies a profound order, a secret language waiting to be deciphered. To study them is to chase the fundamental rhythm of mathematics itself."

"There is another lesser-known "quote" of Erdos that I know first-hand he did not say, since I made it up! ... I gave a talk about Erdos and number theory, and I tried to explain how marvelous the Erdos--Kac theorem is... [Overhead Slide] Einstein: "God does not play dice with the universe." I then said orally: "I would like to think that Erdos and Kac replied..." [Overhead Slide] Erdos and Kac: Maybe so, but something is going on with the primes. ...Somehow, the San Diego newspaper picked this up the next day, and attributed it as a real quote of Erdos."

"Leonhard Euler stated that mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind would never penetrate."

"A prime number is one (which is) measured by a unit alone. Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος."

"And this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long. Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je nappréhendois dêtre trop long."

"In the spring of 1997, Connes went to Princeton to explain his new ideas to the big guns: Bombieri, Selberg and Sarnak. Princeton was still the undisputed Mecca of the Riemann Hypothesis... Selberg had become godfather to the problem... a man who had spent half a century doing battle with the primes. Sarnak... [had] recently joined forces with ... one of the undisputed masters of the mathematics developed by Weil and Grothendieck. Together they had proved that the strange statistics of random drums that we believe describe the zeros in Riemanns landscape are definitely present in the landscapes considered by Weil and Grothendieck. ...It was Katz who, some years before, had found the mistake in Wiless first erroneous proof of Fermats Last Theorem. And finally there was Bombieri... the undisputed master of the Riemann Hypothesis. He had earned his for the most significant result to date about the error between the true number of primes and Gausss guess - a proof of... the Riemann Hypothesis on average. ...Bombieri, like Katz, has a fine eye for detail."

"Riemanns insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. ...[I]n the strange mathematical world beyond Riemanns glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony... would explain why outwardly the primes look so chaotic. The metamorphosis... where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there."

"We dont naturally have some magic formula to try and find these prime numbers. In fact trying to find where the next prime number is represents one of the biggest mysteries in the whole of mathemaics. ...The challenge if the primes is somehow of the ultimate puzzle in the whole of mathematics."

"The Elements contains thirteen books by Euclid, and two, of which it is supposed that Hypsicles and Damascius are the authors. ...The seventh, eighth, ninth books are on the theory of numbers, or on arithmetic. In the ninth book is found the proof to the theorem that the number of primes is infinite."

"(a + b) \times (a - b) = a^2 - b^2...The difference of two square numbers is always a product, and divisible both by the sum and by the difference of the roots of those two squares; consequently the difference of two squares can never be a prime number."

"Prime numbers belong to the exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoys simple, elegant description, yet lead to extreme—one might say unthinkable—complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times... vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes..."