On Numbers: On Prime Numbers (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)

“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)

"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, “Uncle Petros and Goldbach's Conjecture”, 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) 

"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)

“The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, “Mathematics: A Very Short Introduction”, 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)

“Our world resonates with patterns. The waxing and waning of the moon. The changing of the seasons. The microscopic cell structure of all living things have patterns. Perhaps that explains our fascination with prime numbers which are uniquely without pattern. Prime numbers are among the most mysterious phenomena in mathematics.” (Manindra Agrawal, 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)

"Until [the RH is proved], we shall listen enthralled by this unpredictable mathematical music, unable to master its twists and turns. 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. We still await the person whose name will live for ever as the mathematician who made the primes sing." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy,"The Music of the Primes", 2004)

"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)

"The worst aspect of the term 'complex' - one that condemns it to eventual extinction in my opinion - is that it is also applied to structures called 'simple'. Mathematics uses the word 'simple' as a technical term for objects that cannot be 'simplified'. Prime numbers are the kind of thing that might be called 'simple'" (though in their case it is not usually done) because they cannot be written as products of smaller numbers. At any rate, some of the 'simple' structures are built on the complex numbers, so mathematicians are obliged to speak of such things as 'complex simple Lie groups'. This is an embarrassment in a subject that prides itself on consistency, and surely either the word 'simple' or the word 'complex' has to go." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"There are many ways to use unique prime factorization, and it is rightly regarded as a powerful idea in number theory. In fact, it is more powerful than Euclid could have imagined. There are complex numbers that behave like 'integers' and 'primes', and unique prime factorization holds for them as well. Complex integers were first used around 1770 by Euler, who found they have almost magical powers to unlock secrets of ordinary integers. For example, by using numbers of the form a + b√ -2. where a and b are integers, he was able to prove a claim of Fermat that 27 is the only cube that exceeds a square by 2. Euler's results were correct, but partly by good luck. He did not really understand complex 'primes' and their behavior." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Number-theoretic equivalences of the Riemann hypothesis provide a natural method of explaining the hypothesis to nonmathematicians without appealing to complex analysis. While it is unlikely that any of these equivalences will lead directly to a solution, they provide a sense of how intricately the Riemann zeta function is tied to the primes"  (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007

"So the prime number theorem is a relatively weak statement of the fact that an integer has equal probability of having an odd number or an even number of distinct prime factors." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

“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)

"The first and easies proofs [of the prime number theorem] are analytic and exploit the rich connections between number theory and complex analysis. It has resisted trivialization, and no really easy proof is known. This is especially true for the so-called elementary proofs, which use little or no complex analysis, just considerable ingenuity and dexterity. The primes arise sporadically and, apparently, relatively randomly, at least in thes ense that there is no easy way to find a large prime number with no obvious congruences. So even the amount of structure implied by the prime number theorem is initially surprising." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"For both primes and symmetries, zeta functions act as black boxes. They are built from a formula which binds together the numbers you are trying to understand. The hope is that the zeta function will reveal new insights into the numbers of symmetries. It provides a way of getting from part of the mathematical world where chaos seems to reign to a completely different region where one can start to pick out patterns." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. […] If you go on counting, you discover that these pairs gradually become rarer, lost in that silent, measured space made only of ciphers. You develop a distressing presentiment that the pairs encountered up until that point were accidental, that solitude is the true destiny. Then, just when you’re about to surrender, you come across another pair of twins, clutching each other tightly.” (Paolo Giordano, “The Solitude of prime numbers”, 2008)

"Riemann had found a passageway from the familiar world of numbers into a mathematics which would have seemed utterly alien to the Greeks who had studied prime numbers two thousand years before. He had innocently mixed imaginary numbers with his zeta function and discovered, like some mathematical alchemist, the mathematical treasure emerging from this admixture of elements that generations had been searching for. He had crammed his ideas into a ten-page paper, but was fully aware that his ideas would open up radically new vistas on the primes." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Since they represent so natural a sequence, it is almost irresistible to search for patterns among the primes. There are however no genuinely useful formulas for prime numbers. That is to say there is no rule that allows you to generate all prime numbers or even to calculate a sequence that consists entirely of different primes." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008) 

“[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.” (Martin Gardner, “When You Were a Tadpole and I Was a Fish”, 2009)

"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound" (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)