'The Mystery of the Prime Numbers', Secrets of Creation vol. 1 by Matthew Watkins


related curiosities

"Just as practical applications of prime numbers have emerged in the cryptographic, statistical, and other computational fields, there are likewise applications in such disparate domains as engineering, physics, chemistry and biology. Even beyond that, there are amusing anecdotes that collectively signal a certain awareness of primes in a more general, might we say lay, context. Beyond the scientific connections there are what may be called the 'cultural' connections."


R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective (Springer-Verlag, 2001)


  literature          music          graphic art          film          theatre

biology         psychology and consciousness         history         popular

Riemann hypothesis    extra-terrestrial     spirituality/theology     miscellaneous


 
literature

quotes on the primes, the zeta function, and the Riemann hypothesis

more specialised quotes relating to prime numbers and physics
 

Helen Spalding's poem "Let Us Now Praise Prime Numbers"

a hypertext version of Tom Apostol's wonderful poem about the Riemann zeta function

Martin Huxley's limericks describing each individual lecture and discussion at the recent New York workshop on zeta functions and associated Riemann hypotheses!

More limericks by Martin Huxley from number theory conferences at Lille 1997 and Oberwolfach 2000
 

M. Haddon, the curious incident of the dog in the night-time (Doubleday, 2003)

"Prime numbers are also a recurrent theme in the book. Christopher loves primes; he knows each one up to 7,057 and numbers all of this book's chapters with prime numbers. He introduces prime numbers to the reader with a very readable explanation of the Sieve of Eratosthenes, and he makes it clear to even the most uninitiated reader that finding large primes is quite difficult. He also alludes to the Prime Number Theorem (or, rather, his inability to recall it when over-stressed) on p. 212."
(from Maria G. Fung's review for MAA Online)
 

Neal Stephenson - Cryptonomicon. This extraordinary author's latest novel draws heavily on cryptography and also features the Riemann zeta function. Here is an appropriately confusing real-life e-mail exchange betwen Stephenson and the authors of the following paper:

M. Anshel and D. Goldfeld, "Zeta functions, one-way functions and pseudo-random number generators", Duke Mathematical Journal 88 No. 2 (1997) 371-390.
 

Apostolos Doxiadis - Uncle Petros and Goldbach's Conjecture. Like Stephenson's book, this very nice little novel features the Riemann zeta function and includes among its characters a semi-fictionalised Alan Turing (also Hardy, Littlewood, Caratheodory and Godel). As a publicity stunt, publishers Faber have offered $1,000,000 for a proof of the Goldbach Conjecture.
 

C. Sagan, Contact, (Simon & Schuster, 1986). Prime numbers feature in an interstellar transmission received by scientists on Earth.
 

D.L. van Krimpen, Proving the Riemann Hypothesis and other simple things (Lulu, 2007) - a collection of short science-fiction stories, including one called "Juggling with the zeta function".
 

W. Ernst, One Person's Junk (science fiction involving prime numbers in DNA sequences)

In his book River Out of Eden, biologist Richard Dawkins imaginatively llustrates the precision of the DNA code with the story of an imprisoned microbiologist alerting the world to his captors' evil plans by genetically engineering a virus containing a coded message. The message is 'flagged' by a sequence of prime numbers.
 

According to E. Weisstein's MathWorld site,

"Remarkably, noted science fiction author Arthur C. Clarke described the prime spiral in his novel The City and the Stars (1956, Ch. 6, p. 54). Clarke wrote, "Jeserac sat motionless within a whirlpool of numbers. The first thousand primes.... Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed. He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh."

However, Clarke never actually performed this thought experiment (personal communication from E. Pegg Jr., May 27, 2002), thus leaving discovery of the unexpected properties of the prime spiral to Ulam."

More about this, including an explanatory comment from Clarke himself can be found here.
 


music

Number theorist Jeffrey Stopple of UC Santa Barbara has created an audio clip based on the Riemann-von Mangoldt explicit formula. He explains:

"Using the Audio package in Mathematica it is not hard to create a sound file in which each note has the same amplitude and frequency as the corresponding term in the explicit formula, coming from a single zero of the zeta function. In this sound file there are the contributions of the first 100 zeros, added one at a time, in intervals of .2 seconds. Finally all 100 zeros play together for ten seconds. This sound is best listened to with headphones or external speakers. For maximum effect, play it LOUD."

Robert Munafo's Riemann zeta function MP3, an eerie 30 second audio clip based on the zeta function's behaviour on the critical line [documentation]

Andrey Kulsha's Riemann zeta function audio files
 

Chris Caldwell's "The Prime Number Listening Guide"
 

Marcel Langerberger's Prime Music

D. Cummerow's three-part MIDI composition based on the sequence of prime numbers

La Symphonie Primordiale - a prime number based composition by Adrian Bamford

Michael J. Schumacher's The Four Stills

"The Four Stills are expressions of the number system that I have used since 1989 in my electronic compositions. These numbers form the basis for dozens of rhythmic processes executed by the computer, from playing a midi grand piano, to deciding a range within a sample buffer to play, to determining larger structural aspects such as the alternation of sound and silence within a particular part. The fact that they are prime numbers prevents predictable patterns from emerging. The limit of six maintains structural coherence."
 

Joe Monzo's theory of intonation based on prime numbers

A note from sci.math on how the Riemann zeta function relates to tuning.

Further discussion of how Riemann zeta function relates to tuning: [1]   [2]   [3]   [4]

B. Warren, "An interesting group of combination-product sets produces some very nice dissonances", The Journal of the Just Intonation Network 9 (1):1 (1995) 4-9.

"Sets of 12 pitches are generated from a sequence of five consecutive prime numbers, each of which is multiplied by each of the three largest numbers in the sequence. Twelve scales are created in this manner, using the prime sequences up to the set (37, 41, 43, 47, 53). These scales give rise to pleasing dissonances that are exploited in compositions assisted by computer programs as well as in live keyboard improvisations."

J. Dudon, "The golden scale", Pitch I/2 (1987) 1-7.

"The Golden scale is a unique unequal temperament based on the Golden number. The equal temperaments most used, 5, 7, 12, 19, 31, 50, etc. are crystallizations through the numbers of the Fibonacci series, of the same universal Golden scale, based on a geometry of intervals related in Golden proportion. The author provides the ratios and dimensions of its intervals and explains the specific intonation interest of such a cycle of Golden fifths, unfolding into microtonal coincidences with the first five significant prime numbers ratio intervals (3:5:7:11:13)." [Note that here the Fibonacci sequence mentioned differs slightly from, but is closely related to, the usual one.]
 

Musical Theory and Ancient Cosmology (involving primes)
 

Prime numbers and the music of Olivier Messiaen

Messiaen's Quartet for the End of Time

"Messiaen uses rhythmic motifs of 17 notes and 29 notes which play together. The different prime numbers ensure that the motifs are constantly creating new combinations of sound as the music evolves. It gives the piece a sense of timelessness because it takes a very long time before the two motifs will repeat a pattern already heard. Messiaen is using the same principle as the prime number cicada who avoids its periodic predator by choosing a prime number life cycle."
 

M. Rubinstein informed me that he intends to build a stringed musical instrument that allows the exploration of musical relationships of Farey sequences, and that the experimental instrument-maker Harry Partch built a similar one a long time ago. He also informed me that his tabla instructor, Warren Ashford, discovered that Farey sequences are involved in traditional Indian tabla rhythms.
 

John Kaizan Neptune, Prime Numbers

"It just so happens that the traditional Japanese instruments used here all contain prime numbers: 3-stringed shamisen, 5-holed shakuhachi, 13-stringed koto, and 17-stringed bass koto. This also points to a basic fundamental of Japanese arts in general: things are deliberately simplified, often understated, to create a very special kind of space. This is true of traditional Japanese music and the instruments themselves.... we have ten fingers, why only five holes on the shakuhachi? The contemporary music recorded here covers a broad range of textures some of which have definitely not been deliberately simplified. As with much of my music, there are influences from many parts of the world - Japan, Europe, India, and Africa. You can find free rhythm, odd meters, polyrhythm, polyphony, pentatonic and diatonic scales, and even an imitation of African Pygmy yodeling in five. "Simple" instruments made from natural materials, recorded in a natural wood hall, direct to digital 2-track recording, wonderful recording engineers, and good musician friends to share some sounds with.... add it all together and you get Prime Numbers!"
 

Jonas Tauber with William Thomas, Doug Haning, Prime Numbers

"The second in producer Jonas Tauber's Zurich Series for Origin, Prime Numbers is a wholly improvised journey through nine original tunes, all based and inspired by prime numbers. Thoughts of an intellectual exercise are soon laid to rest as the recording unfolds in an organic sea of deeply musical explorations. Featuring longtime purveyors of the Northwest free jazz sound - pianist Doug Haning and drummer William Thomas - Zurich-based bassist Jonas Tauber finds great foils for his impressive, classically rooted virtuosity."
 


graphic art

A. Leatherland - Pulchritudinous Primes

Experimental Prime Visualisations

J.-F. Colonna's graphics inspired by the Riemann zeta function, etc.

T. Armand - Aesthetics of the Prime Sequence

images of the filled Julia set for the Riemann zeta function

some Riemann zeta function wallpaper

"Woman With Prime Numbers" - a painting by Joe Rebholz

"1801: 14 Prime Numbers" - a painting by "faux-primitive" artist Michael Eastman (yours for only $5,750!)

The prime number-inspired artwork of Kenneth A. Huff

Tom Marine's charity painting of the 2000-digit "Millenium prime" discovered by John Cosgrave in 1999. Curiously, this prime was announced to the world on 22/11/99, the day before this website was launched (I had chosen the date 23/11/99 some time earlier, unaware of Cosgrave's discovery). The millenial connection was picked up on by the media, and quite possibly as a result of the sudden burst of coverage, prime numbers were more prominent in popular consciousness on that day than at any other time in history.

"Rationality and Unpredictablility" - and essay on the artist Esther Ferrer

"In many of Esther Ferrer's works, we can perceive rigorous principles of permutation, which she constructs in a rational way. A conscious creation of serial modifications leads to new images. Through it all, the meaning of specific actions is transformed. They become abstract structures in which the usual dimensionality of perception seems diluted. The configurations created in the performances, organised like scores, strangely resemble those created in the paintings in which Esther Ferrer worked with prime numbers.

Those pictures appear as analogies of the deconstructions and reconstructions of the performances. The relationships among the prime numbers become visible through different materials. As the system does not always begin counting from the beginning but for example, from the centre to the sides, it shows constantly changing shapes that are unpredictable. Is there an order behind it all? Esther Ferrer spoke about the creation of a chaos in which she felt there was an internal order."
 

For her 2nd solo show at moniquemeloche, Karen Reimer will exhibit her ongoing series "Endless Set" which made its inaugural debut in early 2007 at VONZWECK. "It is a hard-line conceptual project, based on rules Reimer conceived before threading the first needle. Each crazy-quilt pillowcase is made up of a predetermined number of irregularly shaped, irregularly patterned patches. (Two, the first in the series, is made up of two patches; Thirty-Seven is composed of 37 patches, and so on.) Onto each pillowcase, the artist has stitched the accompanying prime number, a piece of white cotton fabric that is the same height in inches as the number it represents (e.g., Thirty-Seven is 37 inches tall)... As the series progresses and the numbers grow in value and height, the cases become more abstract, the sequence more oblique and indecipherable. The bright patchwork gives in to the growing swathes of white. Any portion of the number that does not fit within the grid of the pillowcase, Reimer has folded back onto itself, obscuring the prime-number sequence even further." [Jake Malooley, Time Out Chicago issue 112]
 

number theory and ancient Chinese aesthetics
 


film

Cube (1997)

"A group of people are trapped in a nightmare lattice of cubic rooms and have to figure out how to escape. Cartesian coordinates and prime numbers play a key role. The moral: factor or die! The most interesting math here is in thinking about how they made the movie."
 

Contact (1997)

"Jodie Foster is perfect when she defines prime numbers for a group of Washington bigwigs and is greeted by blank stares. But why does the movie have to work so hard explaining her devotion to science? The book's nonsense about pi is not in the movie."
 

The Mirror has Two Faces (1996)

"Hunk math prof Jeff Bridges explains the Twin Prime Conjecture (that there are infinitely many pairs of primes only two numbers apart) to dowdy english prof Barbara Streisand who actually gets it. She critiques his calculus teaching. Bridges proposes."
[Remake of Le Miroir a Deux Faces (1959)]
 

A Beautiful Mind (2001)

This is an Oscar-winning "fictionalised biography" about John Nash, whose descent into paranoid schizophrenia was at least partly linked to a compulsion to prove the Riemann Hypothesis. The film contains a scene based around the stream-of-consciousness "proof" which Nash gave at a Columbia University conference in February 1959 (it was this event which fully alerted the mathematical community to his mental illness). Before fleeing the lecture theatre pursued by hallucinated Russian agents, actor Russell Crowe mumbles the memorable line: "The zeros of the Riemann zeta function correspond to singularities in spacetime."
 

more favourable review (M. du Sautoy)           less favourable review (R.Lisker)


Sneakers (1992)

"The number field sieve makes a brief appearance in the Hollywood film Sneakers. Robert Redford sits listening to a young mathematician lecturing about [factorising] very big numbers: 'The number field sieve is the best method currently available. There exists an intriguing possibility of a far more elegant approach...But maybe, just maybe, there is a short cut...' sure enough, this whizz-kid, palyed by Donal Logue, has discovered such a method, 'a breakthrough of Gaussian proportions', and has wired it into a small box which unsurprisingly falls into the evil hands of the film's villain, played by Ben Kingsley. The plot is so outlandish that most viewers must imagine that this could never happen in the real world. Yet as the credits for the film roll, up pops 'Mathematical Advisor: Len Adleman', the A in RSA. As Adleman admits, this is not a scenario that we can guarantee won't happen. Larry Lascar, who wrote Sneakers, Awakening and War Games, came to Adleman to make sure he got the maths right. 'I liked Larry and his desire for verisimilitude, so I agreed. Larry offered money, but I countered with Robert Redford &dnash; I would do the scene if my wife Lori could meet Redford'"

M. du Sautoy, The Music of the Primes (Fourth Estate, 2003) p. 240
 

"To convey an idea of scale: A typical instance of the deepest factoring or primality-proving runs of the modern era involves perhaps 1016 to 1018 machine operations. Similarly, a full-length graphically rendered synthetic movie of today - for example, the 1999 Pixar/Disney movie Toy Story 2 - involves operation counts in the same range. It is amusing that for this kind of Herculean machine effort one may either obtain a single answer (a factor, maybe even a single "prime/composite" decision bit) or create a full-length animated feature whose character is as culturally separate from a one-bit answer as can be. It is interesting that a computational task of say 1017 operations is one ten-millionth of the overall historical computing effort by all Earth-bound machinery."

R.Crandall and C. Pomerance, Prime Numbers: A Computational Perspective (Springer, 2001) p.4
 

The Kubrick/Clarke film 2001 exploited as a dubious metaphor for the mysterious Riemann zeta function
 


theatre

Cicada Dance, a musical/sonic play (involving prime numbers) by Malcolm Ruhl
 

Proof, an award-winning play by David Auburn which revolves around the proof of a result involving prime numbers.
 

"In 2000, an esoteric off-Broadway show called The Five Hysterical Girls Theorem paid homage to [F.N. Cole's 1903] calculation by having one of the girl's [factorise] Cole's number [267 – 1]. Prime numbers are a recurrent theme in this play about a mathematical family's trip to the seaside. The father laments his daughter's coming of age, not because she will be old enough to run off with her lover, but because 17 is a prime number, whereas 18 can be divided by four other numbers!"

M. du Sautoy, The Music of the Primes (Fourth Estate, 2003) p. 224
 


biology

E. Goles, O. Schulz and M. Markus, "Prime Number Selection of Cycles In a Predator-Prey Model", Complexity, 6 No. 4 (2001)

[abstract:]"The fact that some species of cicadas appear every 7, 13, or 17 years and that these periods are prime numbers has been regarded as a coincidence. We found a simple evolutionary predator-prey model that yields prime-periodic preys having cycles predominantly around the observed values. An evolutionary game on a spatial array leads to travelling waves reminiscent of those observed in excitable systems. The model marks an encounter of two seemingly unrelated disciplines: biology and number theory. A restriction to the latter, provides an evolutionary generator of arbitrarily large prime numbers."

"A Biological Generator of Prime Numbers"

Why do periodical cicadas adopt a life cycle which has a prime number of years?

Published scientific literature on periodical cicadas (Magicicada spp.).

J. Yoshimura, "The evolutionary origins of periodical cicadas during ice ages", American Naturalist 149 (1) (1997) 112-124.

[excerpt:] "Periodical cicadas (Magicicada spp.) are known for their strikingly synchronized emergence, strong site tenacity, and unusually long (17- and 13-yr) life cycles for insects. Several explanations have been proposed for the origin and maintenance of synchronization. However, no satisfactory explanations have been made at for the origins of the prime-numbered life-cycles. I present an evolutionary hypothesis of a forced developmental delay due to climate cooling during ice ages. Under this scenario, extremely low adult densities, caused by their extremely long juvenile stages, selected for synchronized emergence and site tenacity because of limited mating opportunities. The prime numbers (13 and 17) were selected for as life cycles because these cycles were least likely to coemerge, hybridize, and break down with other synchronized cycles."
 

J. Tohá and M.A. Soto, "Biochemical identification of prime numbers", Medical Hypotheses, 53 (4) (October 1999) 361-361

[Abstract:] "A biochemical technique is proposed whereby prime numbers may be identified."
 

The following items are all based on the idea that there might be some deep connection between genetic coding and the distribution of prime numbers. They vary in degree of academic credibility, and are not terribly convincing, but perhaps collectively signal an interesting tendency on the level of archetypal psychology

(1) J. Yan, A. Yan, and B. Yan, "Prime numbers and the amino acid code: analogy in coding properties", Journal of Theoretical Biology 151 (3) (1991) 333-341.

[Commentary from R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective (Springer, 2001) p.389:]

"These authors infer that certain amino acid sequences in genetic matter exhibit patterns expected of (binary representations of) prime numbers. In one segment they say:

"Additively generated numbers can be primes or nonprimes. Multiplicatively generated numbers are nonprimes ("composites" in number thoery terminology). Thus, prime numbers are more creative than nonprimes. . . The creativeness and indivisibility of prime numbers leads one to infer that the primes smaller than 64 are the number equivalents of amino acids; or that anmino acids are such Euclid units of living molecules."

The authors go on to suggest Diophantine rules for their theory. The present authors do not intend to critique the interdisciplinary notion that composite numbers somehow contain less information (are less profound) than the primes. Rather, we simply point out that some thought has gone into this connection with genetic codes."

(2) M. Sluyser and E. Sonnhammer , "Molecular biology and futuristic problem solving" (from Visions of the Future, ed. C.A. Pickover, Science Reviews, Northwood, England, 1992, pp. 151-157)

"Sluyser and Sonnhammer take software used to study DNA and RNA and apply it to sequences of prime numbers. RNA sequences involve four bases - A, U, C, G - that bond in pairs to form the famous double helix. Sequences of biological significance are expected to possess some nonrandomness, because patterns in the RNA can cause it to fold into more stable configurations. These have less "free energy" than random unfolded sequences. The article shows that sequences of primes can produce significant results when interpreted as RNA sequences. A side effect is that certain new kinds of regularity in primes become apparent." 

(3) R. Brooks, "Pattern in number...from primes to DNA"

(4) J.-Y. Boulay, "Numeric Structures of the Genetic Code"

[The author explains:] "The number of protons contained in every amino acid and the configuration of DNA's bases of their respective genetic coding are connected by numeric phenomena. These phenomena consist of effects of multiple of prime numbers including the totality of the relations enters the configuration of the genetic code ( 64 codons) and the values of the numbers of protons (or atomic numbers) coded 64 (61 amino acids and 3 stop). These phenomena describe important effects of a symmetry as for their distributions in the board of the genetic code. These phenomena of multiple symmetric imply prime numbers: 7 - 11 - 13."
 

There is in fact an established an established link between DNA and number theory, although rather obscure:

L.A. Newberg and D. Naor, "A lower bound on the number of solutions to the probed partial digest problem", Advances in Applied Mathematics 14 (1993) 172-183.

[abstract:] "The probed partial digestion mapping method partially digests a DNA strand with a restriction enzyme. A probe, which attaches to the DNA between two restriction enzyme cutting sites, is hybridized to the partially digested DNA, and the sizes of fragments to which the probe hybridizes are measured. The objective is to reconstruct the linear order of the restriction enzyme cutting sites from the multiset of measured lengths. In many cases, more than one underlying linear ordering is consistent with a multiset of measured lengths. This article shows that a multiset of N measured lengths can have as many as \Omega(Nt) solutions for any t < \zeta-1(2), where \zeta(t) is the Riemann zeta function and \zeta-1(2) ~ 1.73."

The introductory section of the following paper contains a thorough explanation of the probed partial digest problem (which concerns DNA sequences) and Naor and Newberg's result relating it to the number theoretical function H(n):

B. Chor, P. Lemke and Z. Mador, "On the number of ordered factorizations of natural numbers", Discrete Mathematics 214 (2000) 123-133

[abstract:] "We study the number of ways to factor a natural number n into an ordered product of integers, each factor greater than one, denoted by H(n). This counting function from number theory was shown by Newberg and Naor (Adv. Appl. Math. 14 (1993) 172-183) to be a lower bound on th enumber of solutions to the so-called probed partial digest problem, which arises in the analysis of data from experiments in molecular biology. Hille (Acta Arith. 2 (1) (1936) 134-144) established a relation between H(n) and the Riemann zeta function..."
 

The technique developed here:

B.L. Hao, H.C. Lee and S.Y. Zhang, "Fractal related to long DNA sequences and complete genomes", Chaos, Solitons and Fractals 11 (2000) 825

is used here:

Chung-Ming Ko, "Distribution of the units digit of primes", Chaos, Solitons and Fractals 13 (2002) 1295-1302

[abstract:] "A sequence is formed by the units digit of consecutive prime numbers. The sequence is not random. To visualize the non-randomness of the sequence, we utilize a method put forward by Hao et. al. [Chaos, Solitons and Fractals 11 (2000) 825]. A fractal-like structure is observed."
 

S. Nelson, "A Phyllotaxis Prime Number Sieve"

[Abstract:] "An efficient prime number sieve was found in a daisy integer map. The sieving algorithm is demonstrated for a daisy with Fibonacci (21,34)-phyllotaxis using a three-dimensional diagram of a daisy capitulum and a two-dimensional spreadsheet."
 

V. Bezgin, M. Endo, A. Khrennikov, and M. Yuoko, "Statistical biological models with p-adic stabilization", Dokl. Akad. Nauk 334, no.1 (1994) 5-8.

A. Khrennikov, "p-adic model for population growth", from Fractals in Biology and Medicine, 2, Eds. G.A. Losa, et. al. (Birkhauser, 1998).

A. Khrennikov, "p-adic dynamical systems: description of concurrent struggle in biological population with limited growth", Dokl. Akad. Nauk 361 no. 6 (1998) 752-754.
 


psychology and consciousness

excerpts from Chapter 23 of Oliver Sacks' The Man Who Mistook His Wife for a Hat which documents the astonishing case of a pair of autistic twins who appear to have direct psychic access to the 'landscape' of prime numbers.

In 2005, dubious points were noticed in Oliver Sacks' report. Here is a note by Makoto Yamaguchi of The University of Tokyo.

Exploring similar themes, we find:

M. Anderson, N. O'Connor and B. Hermelin, "A specific calculating ability", Intelligence 26 (1998) 383-403

[abstract:] "We report two experiments that investigate the calculating strategy used by a low IQ savant to identify prime numbers. Hermelin and O'Connor (1990) had suggested previously that this subject may use a procedure first described by Eratosthenes to detect a prime number, namely, dividing a target number by all primes up to the square root of the target number and testing for a remainder. In the first experiment, we compare the reaction times of the savant to decide whether a number is prime with those of a control subject proficient in mathematical calculation. In addition, we measured the savant's speed of information processing using an inspection time task. We found that the reaction times of the savant, although generally faster, followed the same pattern of the control subject who reported using the Eratosthenes procedure. The savant's inspection time indicated that his speed of processing was far superior to that expected from someone of his IQ. In the second experiment, we measured the time it takes mathematics students to divide by different prime numbers and we also tested them on the prime identification task. We used their division times to simulate their performance on the prime number identification task under the assumption that they used the Eratosthenes procedure. We also simulated the reaction times that would result from a simple memory-based procedure for identifying primes. We found that the Eratosthenes simulation, in contrast to the memory simulation, provided a good fit to both the students' and the savant's reaction times. We conclude that the savant is using a complex computational algorithm to identify primes and suggest two explanations of how the apparent contradiction between his low general intelligence and his superior numerical ability might be resolved."

and

A. W. Snyder and D.J. Mitchell, "Is integer arithmetic fundamental to mental processing?: The mind's secret arithmetic"

which references this:

H. Welling, "Prime number identificators in idiot savants: can they calculate them?" Journal of Autism and Developmental Disorder 24 (1994) 199-207.

This Ghent University dissertation is also relevant (although, obviously in Dutch):

P. Butseraen, "Het effect van een dynamische visuele storings-techniek op het identificeren van priemgetallen" ("The effect of a dynamic visual storage technique on the identification of prime numbers")

M. Haddon, the curious incident of the dog in the night-time (Doubleday, 2003)

"Prime numbers are also a recurrent theme in the book. Christopher loves primes; he knows each one up to 7,057 and numbers all of this book's chapters with prime numbers. He introduces prime numbers to the reader with a very readable explanation of the Sieve of Eratosthenes, and he makes it clear to even the most uninitiated reader that finding large primes is quite difficult. He also alludes to the Prime Number Theorem (or, rather, his inability to recall it when over-stressed) on p. 212."
(from Maria G. Fung's review for MAA Online)

Daniel Tammet, a "high-functioning autistic savant" mentions in his book Born on a Blue Day that "I see each prime as a smooth-textured shape, distinct from composite numbers (non-primes) that are grittier and less distinctive. Whenever I identify a number as prime, I get a rush of feeling in my head (in the front center) which is hard to put into words. It's a special feeling, like the sudden sensation of pins and needles.

Sometimes I close my eyes and imagine the first thirty, fifty, hundred numbers as I experience them spatially, synesthetically. Then I can see in my mind's eye just how beautiful and special the primes are by the way they stand out so sharply from the other number shapes. It's exactly for this reason that I look and look and look at them; each one is so different from the one before and the one after. Their loneliness among the other numbers makes them so conspicuous and interesting to me.

There are moments, as I'm falling into sleep at night, that my mind fills suddenly with bright light and all I can see are numbers — hundreds, thousands of them — swimming rapidly over my eyes. The experience is beautiful and soothing to me. Some nights, when I'm having difficulty falling asleep, I imagine myself walking around my numerical landscapes. Then I feel safe and happy. I never feel lost, because the prime number shapes act as signposts."
 

G.G. Globus, "Dual mode ontology and its application to the Riemann Hypothesis", from Brain And Being: At the Boundary Between Science, Philosophy, Language (eds. G.G. Globus, K.H. Pribram, G. Vitiello) (J. Benjamins, 2004) 87-110

[abstract:] "Dual mode quantum brain dynamics (QBD) is reviewed and examined ontologically, with special attention to consciousness, subjectivity and existence. The provenance of dual mode QBD is both ontological and epistemological: presence and trace. Quotidian and monadological ontological interpretations of dual mode QBD are compared. The monadological approach is applied to the Riemann Hypothesis (RH) regarding prime numbers. The prime numbers, Riemann's nontrivial zeros of the zeta function and Being itself are closely connected."
 

a discussion between some Jungian theorists concerning the archetypal, and "poetic/ecstatic" language often found in the literature concerning the fundamental issues surrounding the prime numbers
 

A. Khrennikov, "Human subconscious as a p-adic dynamical system", Journal of Theoretical Biology 193 (1999) 179-196.

S. Albeverio, A. Khrennikov and P. Kloeden, "Memory retrieval as a p-adic dynamical system", Biosystems 49 (1999) 105-115.

D. Dubischar, V. Gundlach, O. Steinkamp, and A. Khrennikov, "A p-adic model for the process of thinking disturbed by physiological and information noise", Journal of Theoretical Biology 197 (1999) 451-467.

D. Dubischar, V.M. Gundlach, O. Steinkamp, and A. Khrennikov, "Attractors of random dynamical systems over p-adic numbers and a model of noisy cognitive processes", Physica D 130 (1999) 1-12

A. Khrennikov, "Learning of p-adic neural networks" (preprint, 1999)

[abstract:] "A p-adic model which describes a large class of neural networks is presented. In this model the staes of neurons are described by digits in the canonical expansion of a p-adic number. Thus each p-adic number represents a configuration of firing and non-firing neurons. We present the algorithm of learning for p-adic neural networks based on the minimization of the error-functional (here we use a random search procedure in the space of p-adic weights). This algorithm (or its more advanced versions) could be applied for image recognition."
 

P.J. Marcer, "A quantum mechanical model of evolution and consciousness" (involving Riemann's zeta function)

I. Miller and R. Marshall, The Auric Key (Syndex Synergetics synopsis) This appears to be a fusion of traditional (e.g. Hindu and Kabbalistic) numerological thought with Buckminster Fuller's 'Synergetics', informed by psychological theorist Carl Jung's work on number symbolism and archetypes. See also "A Synergetic Revisioning of Number Dynamics". This work seems to have been endorsed by Fuller himself, and to be noticeably more 'rigorous' than the vast majority of numerologically-oriented material.

S. Huang, "Prime numbers as lawful creatures of the mind"

prime numbers and messianic delusions

"It's enough to make me conjecture that infinity's prime and Riemann's Zeta function accounts for fractional charge subatomically just for the Higg's boson with an involucral matrix of ogdoad parity as midwife!"

E. Siegel, ""'Primorials' Lurking", or "Primes' Primorials' 'Jump-Chumps' 'Chump-Change'"    [more of Siegel's extraordinary "FUZZYICS" here: 1    2    3    4]

John Nash and the Riemann Hypothesis

a very strange website on John Nash, the Riemann Hypothesis and quantum reality

J.S. Ratcliffe, The Books of Angelhaunt

"In these four Angelhaunt books, Jason Stuart Ratcliff explores his schizophrenia for all the creative treasures it holds within it. Ratcliff puts art prose beyond where it has ever gone, making an epic prose poem, and creating a manifesto to the world of literary art that says, "We don't want tales and stories. Give us pure mind and self, and everything terrible and holy within you.""

Here's part of a summary of one section:

"Lincoln will return, and the world will end. Then, after sixty-one plus one series of sixty-one milleniums of heaven, the world will begin again with all its suffering. Most of their prophets get doctorates in the study of mathematics, and are very intelligent, but dull this intelligence with all kinds of psychedelic substances. They hold that irrational numbers are the cause of all suffering on earth, and that these will not exist in heaven; that the infinite number of irrational numbers increases with time; and that all that existed at the foundation of the world was prime numbers, and that God made the world out of these."
 


history

Prime numbers and African artifact - a note from the sci.anthropology newsgroup

Going to war over prime numbers by Duncan Campbell

Alan Turing's attempts to build a mechanical device for calculating the zeros of the zeta function.

A.R. Booker, "Turing and the Riemann Hypothesis", Notices of the AMS 53 (2006) 1208-1211

Ancient Greeks and Prime Numbers

St. Andrew's University mathematical history project entry for 'prime numbers'
 


popular

Jean-Paul Delahaye, Merveilleux Nombres Premiers - Voyage au Coeur de l'arithmetique     [here is a crude English translation of this French review, courtesy of AltaVista's "Babelfish"]

M. du Sautoy, "The Music of the Primes" - article for popular science readership reproduced from Science Spectra 11 (1998)    [simple text version]

On 28/09/05, BBC4 TV showed, "The Music of the Primes", a documentary presented by du Sautoy.

The BBC Radio 4 programme In Our Time ran an episode on 12/02/06 about prime numbers with three guests, including Marcus du Sautoy.

E. Klarreich, "Prime Time" - another popular exposition from New Scientist (11/11/00)

BBC Radio 4 programme about prime numbers presented by Simon Singh (originally broadcast 29/10/03)

BBC Radio 4 programme Material World (originally broadcast 02/12/04)

Quentin Cooper "talks to Marcus du Sautoy, Professor of Maths at Oxford University and Robin Wilson, Gresham professor of Geometry about the power of primes." (This section of the programme starts about 16 minutes from the beginning.)
 


the riemann hypothesis

K. Sabbagh, "Beautiful Mathematics", Prospect (January 2002)

K. Sabbagh, Dr. Riemann's Zeros: The Search for the $1 Million Solution to the Greatest Problem in Mathematics (Atlantic Books, 2002) - a recently published popular account of the Riemann hypothesis, to be published in the U.S. in April as The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Farrar, Strauss and Giroux)

K. Sabbagh talks to the BBC about writing his book on the Riemann hypothesis

Two more books of a similar nature followed in 2003:

J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, (JHP, 2003)

Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (HaperCollins, 2003)

Observer profile of du Sautoy

Here is K. Leutwyler's comparitive review of all three books from Scientific American.

Here is another, by D. Lim, from The Village Voice.

...and another by J.C. Alexander

A fourth 'popular science' book on the Riemann Hypothesis came out in May 2005:
D. Rockmore, Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers (Random House, 2005)
Here is a radio interview with Dan Rockmore about the book.

The Clay Mathematics Institute offers $1,000,000 for a proof of the Riemann Hypothesis

T. Radford, "Maths holy grail could bring disaster for internet" (from The Guardian, 7th September, 2004)

Professor J.E. Littlewood presents a brief argument as to why he believes the Riemann Hypothesis to be false.

G.G. Globus, "Dual mode ontology and its application to the Riemann Hypothesis", from Brain And Being: At the Boundary Between Science, Philosophy, Language (eds. G.G. Globus, K.H. Pribram, G. Vitiello) (J. Benjamins, 2004) 87-110

[abstract:] "Dual mode quantum brain dynamics (QBD) is reviewed and examined ontologically, with special attention to consciousness, subjectivity and existence. The provenance of dual mode QBD is both ontological and epistemological: presence and trace. Quotidian and monadological ontological interpretations of dual mode QBD are compared. The monadological approach is applied to the Riemann Hypothesis (RH) regarding prime numbers. The prime numbers, Riemann's nontrivial zeros of the zeta function and Being itself are closely connected."

J.R. Partington's humorous article "How I Proved the Riemann Hypothesis" (very British humour...)

D. Zeilberger, The British Government Should Declassify Turing's Counterexample to the Riemann Hypothesis (April 1st, 2000)

What appears to be a sort of prophecy that the RH will be proven by 2003

T. McAlee's theological "proof" of the RH

A. Misra, "Entropy and prime number distribution; (a non-heuristic approach)" includes a philosophical-style argument attempting to prove the Riemann hypothesis, among other things

M. Persuy's semiotic "proof" of the RH   [more of the same sort of thing]

Some proposed proofs of the Riemann Hypothesis (some more serious than others)

"Market odds of "Riemann Hypothesis by 2020" at 65 to 80%"

B. Schechter, "143-year-old problem still has mathematicians guessing". This is a fairly good New York Times article (2 July 2002) on the recent zeta-functions conference at the Courant Institute (02/07/02). The online version requires a user ID and password, but registration is free and only takes a couple of minutes.

Highly tangential and variously zany, philosophical and technical (often wildly inaccurate) discussion inspired by the above article, the Clay prize, etc. can be found at SlashDot.org, the self-described "News for Nerds" site.

J. Derbyshire, "All Things Work for Good", an unpublished magazine article on the Riemann Hypothesis

Andrew Wiles and Arthur Jaffe attempt to explain the Riemann Hypothesis on America's National Public Radio

A.R. Booker, "Turing and the Riemann Hypothesis", Notices of the AMS 53 (2006) 1208-1211

a compass-and-straightedge approach to the Riemann Hypothesis (part of this site)
 


extra-terrestrial

Using Prime Numbers to Communicate with Other Civilizations

"Bipeds and Prime Numbers", a speech by University College Cork's professor of philosophy Garrett Barden

The Arecibo Message - the message consists of 1679 bits, arranged into 73 lines of 23 characters per line (these are both prime numbers, and may help any alien recipients to decode the message).

Drake's Cryptogram

a rather strange proposal to send the Ishango bone into space

C. Sagan, Contact, (Simon & Schuster, 1986). Prime numbers feature in an interstellar transmission received by scientists on Earth.

a well thought-out examination of the possibility that a prime-number based message received from space might not necessarily be due to "intelligent life forms" in the sense we imagine

W. Ernst - One Person's Junk (science fiction involving prime numbers in DNA sequences)

an alt.consipiracy posting claiming that the U.S. National Security Agency has made cryptographic advances as a result of number theory results passed to them by everyone's favourite race of aliens from Zeta Reticuli known as the 'grays'

channelled material, purportedly from the "Cassaeopians", concerning prime numbers

The Tic Xenotation - semi-fictional account by Nick Land, begins by explaining:

"Daniel C. Barker's Tic Xenotation emerged during the highly obscure phase of his life when he was working for 'NASA' (some hesitation is appropriate here) on the SETI-related 'Project Scar' in Southeast Asia, tasked with designing a 'general purpose decryption protocol' for identifying intelligent signal from alien sources."

[part 2]
 


sprirituality and theology

P. Collins, "A Deeper Significance: Resolving the Riemann Hypothesis"
 

I. Mallett, "Does God Think 1 is Prime?"
 

S. Hayes, some stream of consciousness notes on the distribution of primes, consciousness, Taoism and ultimate reality, inspired to some extent by this website

Hayes muses on "the measurable and the immeasurable", primes and some of the ideas on this website (13 minute audio recording)
 

M. Iradier, "The left hand of chaos": This is an intriguing essay in which the author appears to be linking certain aspects of the Riemann zeta function to Ayurveda (about which he is clearly knowledgeable). Although it has already been accepted for publication by an Indian journal, the author is the first to admit that the quality of the English translation is very poor (it was originally written in Spanish). If you would be interested in attempting a better translation, please get in touch with the author.
 

prime numbers in the Quran: [1]   [2]   [3]
 

I. Miller and R. Marshall, The Auric Key (Syndex Synergetics synopsis): This appears to be a fusion of traditional (e.g. Hindu and Kabbalistic) numerological thought with Buckminster Fuller's 'Synergetics', informed by psychological theorist Carl Jung's work on number symbolism and archetypes. See also "A Synergetic Revisioning of Number Dynamics". This work seems to have been endorsed by Fuller himself, and to be noticeably more 'rigorous' than the vast majority of numerologically-oriented material.
 

A Christian blog contemplates the mystery of the prime numbers
 

T. McAlee's theological "proof" of the RH
 

Prime Number 17: A list of multiples of 17 and their mystical associations, provided by the 'Order of Nazorean Essenes'.
 

A curious excerpt found here:

"...there is a growing belief among scientists and mathematicians, a growing faith of religious proportions, that somehow prime numbers are the key to solving the greatest philosophical mystery of all: why there is something instead of nothing."

An even more curious excerpt found here:

"Spiritual Consideration: I cannot write about this yet, but I feel Prime-Numbers are living beings, not as we imagine them as deceased beings for a form and ideas of being human, but they, as all numbers, are conscious beings; and there is a way to address Prime-Numbers in an affirmation and discover their hidden nature which has not been yet discover or cover by the solely mathematical approach. As soon I find reference material or by my own experience I will include this here on this page."

Here's the conclusion to J. Aveleira's "Let It Be: How many pieces are there in a bit of reality?":

"The knowledge we may gather on issues of being, consciousness and reality, possibly is delineable by means of a single, universal, fractal-holographic and evolutionary process based on the unfolding and interaction of unsettled prime numbers of identifiable classes, components or polarities.

Why prime? Because prime numbers are not factorable. In a quest for absolute models, one tends to look for comprehensive, essential concepts, and likely shall find them following structures which would not be reducible into others. Why fractal-holographic? Because the structure of absolute models of reality probably shall surface as a whole in every part, at every instance and scope of actuality. Why evolutionary? Because that process of change appears purposeful, directional, aimed to growth and to the solution of conflicts. Why unsettled? The ultimately detailed and all-encompassing structure of reality possibly shows an undeterminable prime number of classes arranged in unfathomable nuance. Good, veritable representations may show 1 or 2 or 3, 2x2, 5, 2x3, 2x3x2, etc. classes or components and perhaps a little further ones. Those representations may be valid and functional to a remarkable extent. I believe, however, that the infinite complexity possibly extant in ultimate reality would not be touchable by minds subject to any degree of limitation."
 

a discussion between some Jungian theorists concerning the archetypal, and "poetic/ecstatic" language often found in the literature concerning the fundamental issues surrounding the prime numbers
 

number theory and Taoist aesthetics
 

The Kubrick/Clarke film 2001 exploited as a dubious metaphor for the mysterious Riemann zeta function
 

Musical Theory and Ancient Cosmology (involving primes)
 


miscellaneous

Eng. Hatem Al.Bishtawi, "The Orbits of Primes": "Thousands of problems were solved unbelievable outputs were gained. A new revolution in number theory and arithmetic is coming. Practical applications could be expected." Very strange indeed.
 

H. Peter Aleff, Prime Passages to Paradise

"An exploration of patterns in the distribution of primes, and some evidence for their ancient perception. The first volume in this series discusses different ways of organizing the natural numbers to make them reveal some of the order in the distribution of primes."
 

A. Misra, "Entropy and prime number distribution; (a non-heuristic approach)"

This includes philosophical-style arguments attempting to prove the Riemann hypothesis, P vs NP problem, twin primes conjecture and the primality of 1.
 

A. Berezin, "Emergence, Self-Organization and Prime Numbers" [poster from 1998 APS meeting]

[Abstract:] "Pattern of primes (PP) is critical for dynamics of universal emergence, self-organization and complexity ascendance [1-3]. Due to gradual logarithmic dilution of primes (prime number theorem), PP gives only base envelop for above effects. More informative are full factorizational spectra (FS) of all intermediate composites. Tower exponential mappings like f(N) = 10(N)10 with (N) indicating N vertical arrows [4] lead to infinite fractal-like hierarchy of integer trails; say, FS of intervals between f(N) and f(N+1). This allows FAPP-infinite informational content of PP and FS be "used" as catalyzer of emergence dynamics. This is "Platonic pressure effect" (physical embodiments of PP and FS). Said effect may provide more direct picture for cosmogenesis than traditional quantum tunneling ("Big Bang") and/or inflationary scenarios. Furthermore, we can speculate that metrics of (Mega)universe at tower exponential scales becomes asymptotically Euclidean (multi or infinitely dimensional), due to unchangability of PP and FS. - [1] Arnold Arnold, "The Corrupted Sciences", Paladin (Harper Collins), 1992; [2] Peter Plichta, "God's Secret Formula: Deciphering the Riddle of the Universe and the Prime Number Code", Element, 1997; [3] Alexander Berezin, URAM Journal, 20, 72 (1997); [4] Donald E. Knuth, Science, 194, 1235, 17 Dec 1976."

A. Berezin, "Vacuum Fluctuations, Cosmogenesis and Prime Number Gaps" [poster from 2002 APS meeting]

[Abstract:] "Starting from E.Tryon (1973), idea of cosmogenesis through quantum tunnelling "from nothing" became popular. Both complimentary streams of it, inflationary models (Guth, Linde) and quantum parallelism (Everett, Deutsch), require some starting point as, e.g., concretisation of Leibnitz Principle (Omnibus ex nihil decendis sufficit unum). This leads to propositional conjecture (axiom?) that (meta)physical "Platonic Pressure" of infinitude of numbers and Cantor "alephs" becomes an engine for self-generation of physical universe directly out of mathematics: inexhaustibility of Number Theory (NT) drives cosmogenesis. While physics in other quantum branches of inflating universe (Megaverse) can be (arbitrary) different from ours, NT is not (it is unique, absolute, immutable and infinitely resourceful). Energy-time uncertainty principle (UP) allows indefinite lifetime provided we start from total zero energy. Analogue of UP in NT is theorem by H.Maier (1981) stating the existence of arbitrary long trails of isolated primes such that each next gap is arbitrary greater than average gap (logN). On physical level these arbitrary large deviations from Prime Number Theorem translate into permissiveness of (arbitrary) large quantum fluctuations."
 

M. Ryan, New Results on Prime Numbers (self-published book)
 

H. Müller, "The physics of the number line"
This can be found as part of a page outlining Müller's "global-scaling" theory-of-everything. He believes that the distribution of matter in the Universe can be related to the distribution of primes.

"The world of scales is nothing else but the logarithmic line of numbers known to mathematics at least since the time of Napier (1600). What is new, however, is the fundamental recognition that the number line has a harmonic structure which is itself the cause for the standing pressure wave."

An article covering the same material can be found in Nexus magazine vol 11, no. 5 (August-September 2004)
 

This comes from an alt.sci.physics.new-theories posting by El Hemetis:

"A zero is significant when it is a Riemannian coordinate that participates in a topologically invariant feature of a surface, which I happen to stumble on as that elusive centerline of a Mobius strip. The line Re (s) = 1/2 only at the middle of a regular Mobius strip, on which all significant zeros fall demonstrating that deviating from that line would not split the strip preserving the continuity of the orbit. If you relate my work on prime numbers and their relation to TKTODO of my Quantology you shall understand where I am coming from (mathematically speaking). While the relation between a Möbius strip and the zeta function is crystal clear to me, it seems to elude many whom I have discussed it with and I gave up."
 

The (Barkerian) Tic Xenotation "provides a numerical semiotic adapted to the Naturals with special affinity to Euclid's Fundamental Theorem of Arithmetic. The TX constructs numbers in terms of their basic arithmetical features as primes or composites in a notation without modulus (base), place-value or numerals."

[here is Nick Land's earlier, introductory, posting from the same forum]
 

R.L. Bagula, "Prime metasymmetry and inside out disk annular geometry" (16/01/99).

R.L. Bagula, "The Information in the prime sequence and new chaos" (03/10/01)
 

Irish teenager David Doherty has won the 2002 ESAT Young Scientist of the Year Award by proving a conjecture related to the distribution of primes. His continuing research project entitled "The Distribution of the Primes and the Underlying Order to Chaos" seeks to investigate parallels with thermodynamics, entropy, etc.
 

the mysterious Ulam spiral phenomenon
 

S. Chambers presents A New Definition of Prime Number
 

a page from Stephen Wolfram's A New Kind of Science which demonstrates how cellular automata could be used to compute primes
 

how to build a 'Prime-O-Graph'
 

Butterfly Primes
 

illegal primes
 

William (Larry) and Nha Trang Pensinger's extraordinary, dense writings on the Riemann Hypothesis and zeta function, Post's m-valued logics, Charles Muses' hypernumbers and chronotopology, Gödel, Kabbalah, Sanskrit, "laser Esperanto", superconductant DNA, neurophysics, Kant, Schönberg, Kandinsky, psychoneurotic posturing, the Nazi invasion of Poland, non-orientable self-reference and synaesthesia.

Larry Pensinger's immediate reaction to Karpenko's work relating m-valued logics and prime numbers
 

Some theorems on the Riemann zeta function by El'Melik (a purported time-traveller from 3120 A.D.?), with a $999 cash prize for disproof! It seems that these theorems are claimed to be from the 4th millenium, and presented as evidence for the reality of time-travel. No proof of the Riemann hypothesis is included, so we may have to wait another 1200 year at least...
 

a prime-generated pattern supposedly resembling an astronomical radiant
 

The Prime Number Maze
 

Bizarre Finnish humour: Alkulukuja Paskova Karhu - The Prime Number Sh**ting Bear

Kermit the Riemann zeta function Krab (?) - scroll down the page a bit...
 

Feel the power of the Riemann zeta-function!
 

Rejoice! Someone has found another 'big' prime number.




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