Number Theory Quotes

“Some numbers, even large ones, have no factors - except themselves, of course, and 1. These are called prime numbers, because everything they are starts with themselves. They are original, gnarled, unpredictable, the freaks of the number world.”
― Richard Friedberg

“It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.”
― Adrien-Marie Legendre

“Mathematicians call it “the arithmetic of congruences.” You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, … up to 11. If the time is eight o’clock, and you add 9 hours, what do you get? Well, you get five o’clock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 ≡ 5 (mod 12), pronounced “eight plus nine is congruent to five, modulo twelve.”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“No part of mathematics is ever, in the long run, "useless." Most of number theory has very few "practical" applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.”
― C. Stanley Ogilvy, Excursions in Number Theory

“It is shown that the golden ratio plays a prominent role in the dimensions of all objects which exhibit five-fold symmetry. It is also showed that among the irrational numbers, the golden ratio is the most irrational and, as a result, has unique applications in number theory, search algorithms, the minimization of functions, network theory, the atomic structure of certain materials and the growth of biological organisms.”
― Richard A. Dunlap, GOLDEN RATIO AND FIBONACCI NUMBERS, THE

“If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good; and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.”
― G.H. Hardy, A Mathematician's Apology

“{W}e can transfer the study of any formal system—in fact the study of ”
― Douglas R. Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid

“[W]e can transfer the study of any formal system—in fact the study of [all] formal systems—into number theory.”
― Douglas R. Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid