Prime Numbers III

“One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the ‘ideal gases”’which physicists are so fond of. Considered from an external point of view, the distribution is - in broad terms - deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted.” (Gerald Tenenbaum & Michael M France, “The Prime Numbers and Their Distribution”, 2000)

“"As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." (Gerald Tenenbaum & Michael M France, “The Prime Numbers and Their Distribution”, 2000)

“Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy 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, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall and Carl Pomerance, “PrimeNumbers: A Computational Perspective”, 2001)

“The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so."(Timothy Gowers, “Mathematics: A Very Short Introduction”, 2002)

“The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes.”  (David Ruelle, “The Mathematician's Brain”, 2007)

“Although the prime numbers are rigidly determined, they somehow feel like experimental data." Timothy Gowers, “Mathematics: A Very Short Introduction”, 2002)

“[Primes] are full of surprises and very mysterious […] They are like things you can touch. […][ In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles.” (Yoichi Motohashi, “The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics”, 2002)

“[…] despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge.” (Marcus du Sautoy, “The Music of the Primes”, 2003)

“The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle.” (Marcus du Sautoy, “The Music of the Primes”, 2003)