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Prime Numbers


127 = 2^7 - 1 is a Mersenne prime, called M7.

65,537 = 2^2^4 + 1 is the largest known Fermat prime. It is called F4, but is not likely to be confused with the exceptional Lie algebra F4.

2,147,483,647 = 2^31 - 1 is a Mersenne prime. It was shown to be prime by Euler. It is called M31, but is not likely to be confused with the Andromeda galaxy M31. (see Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin, 1986)

The Mersenne prime 2^859433-1 (258716 digits) found by Slowinski and Gage in 1994.

For a number of the form 2^p + 1 to be prime, p must be of the form 2^k.

The Fermat primes, of the form 2^2^k + 1, include:

The Mersenne Primes, of the form 2^k - 1 for prime k, include:

Some other Mersenne Primes are 2^k - 1 for k = 5, 13, 17, 19, 31, 61, 89, 107, 521, 607, and 1279.

Mersenne Numbers that are not prime include 2^11 - 1 = 23 x 89.


Perfect Numbers

are described in a web article by J. J. O'Connor and E. F. Robertson:

"... Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number. ... The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.
6 = 1 + 2 + 3, 
28 = 1 + 2 + 4 + 7 + 14, 
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 

... If, for some k > 1, 2^k - 1 is prime then 2^(k-1)(2^k - 1) is a perfect number. ... Primes of the form 2^p - 1 are called Mersenne primes. ... 2^127 - 1 is a Mersenne prime and so 2^126(2127- 1) is indeed a perfect number. ... The problem of whether an odd perfect number exists ... remains unsolved. ... Today 39 perfect numbers are known, 2^88(2^89- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer. In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers. At the moment the largest known Mersenne prime is 2^13466917 - 1 (which is also the largest known prime) and the corresponding largest known perfect number is 2^13466916(2^13466917 - 1). It was discovered in December 2001 and this, the 39th such prime to be discovered, contains more than 4 million digits. ... Also worth noting is the fact that although this is the 39th to be discovered, it may not be the 39th perfect number as not all smaller cases have been ruled out. ...".


In addition to the rational prime numbers over the reals R, 
you can look at prime numbers over the other division algebras.
Integral complex numbers whose (square) norm is rational prime, 
such as 2 + i with (square) norm 4+1 = 5, 

prime complex numbers.

Depending on which integral complex lattice you use, 
there exist some additional prime numbers.  

The Gaussian integers are G = { a + ib : a,b in Z, 
  i = sqrt(-1)  which is a fourth root of unity } 
the Eisenstein integers are E = { a + wb : a,b in Z, 
  w = ( -1 + i sqrt(3) ) / 2  which is a cube root of unity) }
as described in Conway and Sloane Sphere Packings, Lattices, and Groups (3rd edition, Springer, 1999), in chapter 2, section 72.6, pages 52-55).
Gaussian primes (based on 1 and i) are: 
       1 + i   and its associates; 
       rational primes  4n + 3  and their associates; 
       factors  a + bi  of rational primes  4n + 1  
Eisenstein primes (based on 1 and w = a cube root of 1) are: 
       1 - w   and its associates; 
       rational primes  3n + 2  and their associates; 
       factors  a + bw  of rational primes  3n + 1   
Gaussian and Eisenstein primes have unique factorization, 
up to order, units, and associates.  
(See Guy, Unsolved Problems in Number Theory, 2nd ed, Springer-Verlag 1994)
Both the complex Gaussian integers 
and the complex Eisenstein integers 
are contained in the 4-dimensional integral quaternions as 
2-dimensional sections.  


Integral quaternions whose (square) norm is rational prime, are prime quaternions, and there are no other prime quaternions. In particular, over the quaternions, no rational primes are prime.

I think that integral octonion Cayley numbers are also prime if and only if their (square) norm is rational prime.

Hardy and Wright, in their book on Number Theory, 
use quaternion primes to prove the 4-square theorem 
(that all positive integers are the sum of 4 squares), 
which follows from the proposition 
that all rational primes are the sum of 4 squares, 
because the product of two sums of 4 squares is a sum of 4 squares.  
As Hurwitz proved, 
for the product of two sums of N squares to be a sum of N squares, 
N must be 1, 2 ,4 or 8.  
(Cayley discovered the octonions while 
trying to find the formula for 8 squares.)  

  You can also define

primes over algebraic extensions

of the rational integers, 
such as  the set of   a + b t   where t = (1/2)(1 + sqrt(5)), 
for rational integers a and b.  
Here the primes are: 
       sqrt(5)   and its associates; 
       rational primes   5n  +/-  2   and their associates; 
       factors  a + bi  of rational primes   5n  +/-  1  
Hardy and Wright use such algebraic extensions, 
by sqrt(5) and sqrt(3), to prove primality of some Mersenne primes.  
Conway and Sloane use algebraic extension 
of quaternions by the Golden (1/2)(1 + sqrt(5)) 
to construct the 8-dimensional E8 lattice 
and the 24-dimensional Leech lattice.  

  The pattern of rational prime numbers is interesting. For example, consider the

Riemann zeta function

which is related to Bernoulli Numbers


whose zeroes are shown on a web page of

zeta(s)  is the analytic continuation over the Complex Plane 
         of the sum over N from 1 to infinity SUM( 1 / N^s  ) 
         which sum converges over the Real axis for s > 1. 
         zeta(s) has a pole at s = 1.    

It is also equal to 
zeta(s)  =  PROD(  1 / (1 - P^(-s))   ) 
         product over all prime numbers P 

which is equivalent to 
zeta(s)  =  PROD(  P^s / ( P^s - 1 )  ) 
         product over all prime numbers P.  

Note that the denominators can be written as P^s -1, 
including, for example, 2^s - 1 which is the form 
of the Mersenne Primes. 

Note that zeta(1)  = SUM(1/N)  is the harmonic series. 
The fact that the harmonic series diverges 
shows that 
the sum over all primes P    SUM( 1/P )   also diverges, 
which also shows that the number of primes is infinite.  

(There is a theorem that if PROD( 1 + K ) converges, 
then SUM( K ) converges.) 

(see Introduction to Calculus and Analysis, 
vol. 1, by Courant and John, Springer 1989)  

You can also use zeta functions and generalizations 
to calculate distributions of prime numbers, and 
to do calculations for sum-over-histories path
integrals in quantum theory, and 
for a lot of neat poorly understood stuff.  
According to the 18 May 1996 issue of the New Scientist, 
(see also Science 274 (20 Dec 96) 2014-2015) 
Michael Berry and Jon Keating have seen correspondences 
between the spacing of the prime numbers 
and the spacing of energy levels 
of quantum systems that classically would be chaotic.  
They would like to find a chaotic system that, 
when quantized, would have energy levels that are 
distributed exactly as the prime numbers.  
Since energy levels are positive numbers, 
and so should correspond to a straight line in the complex plane, 
such a zeta function - quantum system correspondence 
could be used to verify the Riemann hypothesis, 
that all the nontrivial zeroes of the zeta function 
are on the straight line  Re(z) = 1/2  in the complex plane. 
Thus, the quantum harmonies in the music of the primes 
could prove the Riemann hypothesis.  

The Berry-Keating systems are not the only systems 
that might be related to the Riemann zeta function,
and similar functions, 
and the idea of using quantum systems to prove 
the Riemann hypothesis did not originate with them. 

Alain Connes, in math.NT/9811068, said:

"... It is an old idea, due to Polya and Hilbert that in order to understand the location of the zeros of the Riemann zeta function, one should find a Hilbert space H and an operator D in H whose spectrum is given by the non trivial zeros of the zeta function. The hope then is that suitable selfadjointness properties of D (of i ( D - 1/2 ) more precisely) or positivity properties of DELTA = D(1 - D) will be easier to handle than the original conjecture. The main reasons why this idea should be taken seriously are first the work of A. Selberg in which a suitable Laplacian is related in the above way to an analogue of the zeta function, and secondly the theoretical and experimental evidence on the fluctuations of the spacing between consecutive zeros of zeta. ... We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of Adele classes. This reduces the Riemann hypothesis to the validity of the trace formula ...
  • Let us first describe ... the direct atempt to construct the Polya-Hilbert space from quantization of a classical dynamical system. The original motivation for the theory of random matrices comes from quantum mechanics. In this theory the quantization of the classical dynamical system given by the phase space X and hamiltonian h gives rise to a Hilbert space H and a selfadjoint operator H whose spectrum is the essential physical observable of the system. For complicated systems the only useful information about this spectrum is that, while the average part of the counting function, (1) N(E) = # eigenvalues of H in [0, E] is computed by a semiclassical approximation mainly as a volume in phase space, the oscillatory part, (2) Nosc(E) = N(E) - <N(E)> is the same as for a random matrix, governed by the statistic dictated by the symmetries of the system. ... the oscillatory part Nosc(E) behaves in the same way as for a random real symmetric matrix ...[and]... gives very precious information on the hypothetical "Riemann flow" whose quantization should produce the Polya-Hilbert space. The point is that the Euler product formula for the zeta function yields a similar asymptotic formula for Nosc(E) ... The Riemann flow cannot satisfy time reversal symmetry. However there are two important mismatches between the two formulas ... The first one is the overall minus sign ... the second one is that ... we do not have an equality for finite values of m ...
  • The basic properties of the Riemann zeta function extend to zeta functions associated to an arbitrary global field, and it is unliquely that one can settle the problem of the spectral interpretation of the zeros, let alone find the Riemann flow, for the particular case of the global field Q of rational numbers without at the same time settling these problems for all global fields. The conceptual definition of such fields k, is the following: A field k is a global field iff it is discrete and cocompact in a (non discrete) locally compact semisimple abelian ring A. As it turns out A then depends functorially on k and is called the Adele ring of k, often denoted by kA. Thus though the field k itself has no interesting topology, there is a canonical and highly non trivial topological ring which is canonically associated to k. When the characteristic p of a global field k is > 0, the field k is the function field of a non singular algebraic curve S defined over a finite field Fq included in k as its maximal finite subfield, called the field of constants. One can then apply the ideas of algebraic geometry, first developed over C , to the geometry of the curve S and obtain a geometric interpretation of the basic properties of the zeta function of k ... The minus sign which was problematic in the above discussion admits here a beautiful resolution since the analogue of the Polya-Hilbert space is given ... by the cohomology group ... which appears with an overall minus sign in the Lefchetz formula ... For the general case this suggests ... The Polya-Hilbert space H should appear from its negative ... In other words, the spectral interpretation of the zeros of the Riemann zeta function should be as an absorption spectrum rather than as an emission spectrum ...
  • There is a third approach to the problem of the zeros of the Riemann zeta function, due to G. Polya and M. Kac ... It is based on statistical mechanics and the construction of a quantum statistical system whose partition function is the Riemann zeta function. Such a system ... does indicate using the ...[correspondence of]... Spectral interpretation of the zeroes ...[with]... Eigenvalues of action of Frobenius on l-adic cohomology ... (namely the correspondence between quotient spaces and noncommutative algebras) what the space X should be in general: (1) X = A / k* namely the quotient of the space A of adeles, A = k_A by the action of the multiplicative group k* ... This space X already appears in a very implicit manner in the work of Tate and Iwasawa on the functional equation. It is a noncommutative space in that, even at the level of measure theory, it is a tricky quotient space. For instance at the measure theory level, the corresponding von Neumann algebra, (3) R01 = Linfinity(A) X| k* where A is endowed with its Haar measure as an additive group, is the hyperfinite factor of type IIinfinity ...[according to Week 175 by John Baez, "... There is more than one type IIinfinity factor, but ... there is only one that is hyperfinite. You can get this by tensoring the type Iinfinity factor and the hyperfinite II1 factor. ... every type In factor is isomorphic to the algebra of n x n matrices. Also, every type Iinfinity factor is isomorphic to the algebra of all bounded operators on a Hilbert space of countably infinite dimension. ..."]... There is a close analogy between the construction of the Hilbert space ... and the construction of the physical Hilbert space ... in constructive quantum field theory, in the case of gauge theories ... Our construction of the Polya-Hilbert space bears some resemblance to ... D. Zagier, Eisenstein series and the Riemannian zeta function, Automorphic Forms, Representation Theory and Arithmetic, Tata, Bombay (1979), 275-301. ...".


Nicholas M. Katz and Peter Sarnak, in Bulletin (NS) AMS 36 (1999) 1-26, say:

"... Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing.
  • Firstly, there are the "function field" analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology.
  • Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection.
  • Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. ...

... During the years 1980-present, Odlyzko has made an extensive and profound numerical study of the zeroes and in particular their local spacings. He finds that they obey the laws for the (scaled) spacings between the eigenvalues of  a typical large unitary matrix. That is they obey the laws of the Gaussian (or equivalently, Circular) Unitary Ensemble GUE ... At the phenomenological level this is perhaps the most striking discovery about zeta since Riemann. The big questions, which we attempt to answer here, are, why is this so and what does it tell us about the nature (e.g. spectral) of the zeroes? Also, what is the symmetry behind this "GUE" law? Odlyzko's computations were inspired by the 1974 discovery of Montgomery that the "pair correlation" of the zeroes is, at least for a restricted class of test functions, equal to the GUE pair correlation R2 ... The Riemann Zeta Function is but the first of a zoo of zeta and L-functions for which we can ask similar questions. ...

... A classical and concrete example of a form on GL2 / Q is f = delta :

(15) delta(q) := q PROD( 1<n<infinity) ( 1 - q^n)^24 =

= SUM( 1<n<infinity) tau(n) q^n

With q = exp( 2 pi i z ) , delta(z) is a holomorphic cusp form of weight 12 for GAMMA = SL2( Z ). That is for z ...[in]... the upper half plane, we have

delta( (az+b)/(cz+d) ) = (cz+d)^12 delta(z) ,

for all a,b,c,d in Z, ad-bc = 1. Its L-function is

(16) L(s,delta) = SUM( 1<n<infinity) ( tau(n) / n^(11/2) ) n^(-s)

... and it is entire ...

... In the 50's ... Wigner suggested that the resonance lines of a heavy nucleus (their determination by analytic means being intractable) might be modeled by the spectrum of a large random matrix. To this end he considered various ensembles (i.e. probability distributions) on spaces of matrices: in particular, the Gaussian Orthogonal Ensemble, 'GOE', and Gaussian Unitary Ensemble, 'GUE'. These live on the linear space of real symmetric (resp. hermitian) N x N matrices and are orthogonal (resp. unitary) invariant ensembles. ... Later Dyson introduced his three closely related circular ensembles: COE, CUE, as well as CSE with it's associated Gaussian Symplectic Ensemble, 'GSE'. These circular ensembles may be realized as the compact Riemannian symmetric spaces (with their volume form as probability measure) U(N) / O(N), U(N) andU(2N) / USp(2N),respectively. He investigated the local spacing statistics for the eigenvalues of the matrices in these ensembles ... in their standard realization ... He shows that these statistics agree with the corresponding matrices from the GOE, GUE and GSE ensembles. ...

... For our purposes of symmetry associated with L-functions, only the first 4 ensembles in Table 1 will play a role. These 4 are the classical compact groups which with a bi-invariant metric yield the so-called type II symmetric spaces ... The invariant volume form on G(N) is just Haar measure. ... some of ...[the]... various families ... are as follows ...

  • I. The family F of Dirichlet L-functions L(s,chi), chi(primitive) of conductor c_chi = q and chi quadratic (i.e. chi^2 = 1) ... The discussion of the function field analogue suggests that G( F ) =Sp. ... The first person to compute numerically the zeroes of L(s,chi) in this family appears to be Hazelgrave. He found that the zeroes "repel" the point s = 1/2, and this is referred to as the Hazelgrave phenomenon. ... One can carry out all of the above for L(s,chi), chi^3 = 1 (or any other order bigger than 2). One finds a unitary symmetry and the zeroes do not repel s = 1/2. ... As remarked in Section 2 the density nu_1(Sp) vanishes to second order at s = 0. Thus, the Hazelgrave phenomenon is a manifestation of the symplectic symmetry! ...
  • II. The family delta x chi [tensor product where delta is the weight 12-cusp form in (15) and chi runs over the quadratic characters. ... From function field considerations we expect G(F) = Owith the refinement that the sub-family F+ with epsilon = 1 has an SO(even) symmetry and F- with epsilon = -1 an SO(odd) symmetry. ...
  • III. The family H(N) of holomorphic Hecke-eigen forms of weight 2 ... The expected symmetry G(F) is O, though as yet we have not understood the function field analogue. ...
  • IV. The family of symmetric-square L-functions ... we expect G(F) =Sp. ...

... Phenomenologically, it is found that the distribution of the high zeroes of any L-function follow the universal GUE Laws, while the distribution of the low-lying zeroes of certain families follow the laws dictated by symmetries associated with the family. The function field analogues of these phenomena can be established, and the source of the symmetry is the monodromy of the family and its scaling limits. Analytic results concerning the distribution of high zeroes for an individual L-function and low zeroes of a family of L-functions, to the extent to which these can be established, confirm these findings above. Whether in the case of L-functions (over Q ) there is indeed some kind of underlying monodromy group which glues the family and is the source of the symmetry is a fascinating question. Our belief is that there is. One can imagine that to each L(s; f), f in F , there is a natural interpretation of the zeroes of L(s; f) as the eigenvalues of an operator U(f) on some space H. As f varies over F these U(f)'s become equi-distributed in the space of such operators with a given symmetry type. For the families discussed ...[above].... these symmetries are identified. In particular, the Riemann Zeta function sits in Family I ... which has a symplectic symmetry. We infer that in the proposed spectral interpretation of the zeroes of the Riemann Zeta function, the operator should preserve a symplectic form! ... We believe that the further understanding of the source of such symmetries holds the key to finding a natural spectral interpretation of the zeroes. ...". 


Some speculative thoughts about John Nash and the Riemann Hypothesis:

In the John Nash biography A Beautiful Mind Sylvia Nasar says, at pages 215-221:

"... Nash would blame ... his attempt to resolve contradictions in quantum theory, on which he embarked in the summer of 1957 ... for triggering his mental illness ... The Institute for Advanced Study ... on Princeton's fringes ... By 1956, Einstein was dead, Goedel was no longer active, and von Neumann lay dying in Bethesda. Oppenheimer was still director ... The Institute was about the dullest place you could find ... Nash was soon spending at least as much time ...[at]... the Courant Institute of Mathematical Sciences at New York University ... as at the Institute for Advanced Study ... Nash left the Institute for Advanced Study on a fractious note. In early July he apparently had a serious argument with Oppenheimer about quantum theory ... Nash's letter ... to Oppenheimer provides the only record of what he was thinking at the time. Nash ... wrote ... "I want to find a different and more satisfying under-picture of a non-observable reality ... most physicists (also some mathematicians who have studied Quantum Theory) ...[are]... quite too dogmatic in their attitudes ...[and tend to treat]... anyone with any sort of questioning attitude or a belief in "hidden parameters" ... as stupid or at best a quite ignorant person.". ...".

To understand the context of having Oppenheimer as a boss while you are trying to work on quantum theory, here is a 1951-52 quote of Oppenheimer from The Bohm biography Infinite Potential, by F. David Peat (Addison-Wesley 1997),page 133: "if we cannot disprove Bohm, then we must agree to ignore him."

It is also possible that Nash's work on the Riemann Zeta function might have been related to some ideas about quantum theory, along lines suggested by Hilbert and Polya, which lines of thought (not very fashionable in the 1950s) have now become very respectable,

so I speculate that Nash might have had a valid insight about connection between the Riemann zeta function and Bohm-type quantum theory, and that Oppenheimer et al, who hated Bohm, would have shaken Nash by their hostility to such ideas, and that a math/physics audience at the time would not have been able to appreciate any such connection.

In the movie A Beautiful Mind, Nash was depicted as associating the Riemann zeta zeroes with spacetime singularities, whereupon the audience began to get confused and Nash ran off-stage. Maybe in real life Nash had been associating quantum levels with the zeroes, which might in fact be true, but might well have seemed to be incomprehensible gibberish to a 1950s audience.



Some recent (as of July 2003) developments with respect to Proof of the Riemann Hypothesis include:

Louis de Branges de Bourcia,

who proved the Bieberbach conjecture but had great difficulty in getting his proof accepted by USA mathematicians (He had to go to the USSR to get people to read it seriously, but there is no more USSR so that method of getting something accepted is no longer available), has written a paper (latest version 15 November 2002) entitled Riemann Zeta Functions in which he says: "... A Riemann zeta function is a function which is analytic in the complex plane, with the possible exception of a simple pole at one, and which is characterized by an Euler product and a functional identity. Riemann zeta functions originate in an adelic generalization of the Laplace transformation which is defined using a theta function. Hilbert spaces,whose elements are entire functions, are obtained by application of the Mellin transformation. Maximal dissipative transformations are constructed in these spaces which have implications for zeros of zeta functions. The zeros of a Riemann zeta function in the critical strip are simple and lie on the critical line. The Euler zeta function, the Dirichlet zeta functions, and the Hecke zeta functions are examples of Riemann zeta functions. ... Hypotheses were ... on a space H(E) ... An entire function Q(z) is given which is associated with the space. ... The more general choice of Q(z) is essential to the proof of the Riemann hypothesis for Riemann zeta functions which are analytic in the complex plane. ... A variant of these hypotheses is applied in the proof of the Riemann hypothesis for the Riemann zeta function which is not analytic in the complex plane. ... A proof of the Riemann hypothesis for the zeta function of order v and character X for the adelic plane is obtained from the maximal dissipative transformations in the Sonine spaces of order v and character X for the adelic plane when the zeta function is analytic in the complex plane. ... A proof of the Riemann hypothesis for the zeta function of zero order and principal character for the adelic plane is obtained from the maximal transformations of dissipative deficiency at most one in the augmented Sonine spaces of zero order and principal character for the adelic plane. ... A proof of the Riemann hypothesis for the zeta function of order 2v + 1 and character X for the adelic diplane is obtained from the maximal dissipative transformations in the Sonine spaces of order 2v + 1 and character X for the adelic diplane. ...".

In a document on his web site named apology.pdf, de Branges said:

"... The research project on the Riemann hypothesis has received such opposition in the mathematical community as to make publication in journals and participation in symposia difficult. ... The current status of the work on the Riemann hypothesis is available electronically as a preprint of Purdue University. The existing manuscript should be sufficient evidence that the conjecture is true. The manuscript will be revised until the clarity and completeness of the argument convinces the mathematical community. ...".

That document, apology.pdf, was revised and the revised document contains the date 18 March 2003. The new version can now be found on the web here, according to a link on the web page of de Branges. The new version is longer (23 pages) than the old one (9 pages), but the new version does not contain the above-quoted language. The new version, which seems to me to be more detailed/historical in tone, is entitled APOLOGY FOR THE PROOF OF THE RIEMANN HYPOTHESIS, and it concludes with the following paragraph:

"... A curious coincidence needs to be mentioned as part of the chain of events which concluded in the proof of the Riemann hypothesis. The feudal family de Branges originates in a crusader who died in 1199 leaving an emblem of three swords hanging over three coins, surmounted by the traditional crown designating a count, and inscribed with the motto "Nec vi nec numero." This is a citation from Chapter 4, Verse 6, of the Book of Zechariah: "Not by might, nor by power, but by my Spirit, says the Lord of Hosts." The chateau de Branges was destroyed in 1478 by the army of Louix XI of France during an unsuccessful campaign to wrest Franche-Comte from the heirs of Charles the Bold of Burgundy. The family de Branges performed administrative, legal, and religious functions in Saint-Amour for the marquisat d'Andelot during Spanish rule of Franche-Comte. Francois de Branges of Saint-Amour received the seigneurie de Bourcia in 1679 when Franche-Comte became part of France. The chateau de Bourcia remained the home of his descendants until it was destroyed by Parisian revolutionaries in 1791. The chateau d'Andelot near Saint-Amour, which survived the revolution, was bought in 1926 by Pierre du Pont, an elder brother of Irenee du Pont, for a nephew assigned in diplomatic service to France. This coincidence accounts for the interest which Irenee du Pont showed in a student of mathematics. The ruin of the chateau de Bourcia overlooks a fertile valley surrounded by wooded hills. The site is ideal for a mathematical research institute. The restoration of the chateau for that purpose would be an appropriate use of the million dollars offered for a proof of the Riemann hypothesis. ...".

Note - It is my opinion that de Branges uses the word "apology" in its French meaning, which is (according to Mansion's Shorter French and English Dictionary):

"... apologie ... Defence; vindication, (written) justification (de, of) ... to vindicate, justify, defend, s.o. NOTE. Never = EXCUSE ...".

It is my opinion that de Branges did NOT intend that his use of the word "apology" should be interpreted with the American/English number 1. meaning (according to The American Heritage Dictionary of the English Language):

"... apology ... 1. An acknowledgement expressing regret or asking pardon for a fault or offense. ...".


The American Institute of Mathematics sponsored a Conference

around July 2002 on the Riemann Hypothesis. According to a New York Times article by Bruce Schechter about the conference: "... Over the past few decades billions of zeros of the zeta function have been calculated by computer, and every one of them obeys Riemann's hypothesis. ... But the field is rife with examples of hypotheses that seem to be true but are subsequently proven to fail at numbers beyond the reach of any conceivable computer. Only a mathematical proof, based on logic, can handle questions of the infinite. Still, calculating the zeros of zeta is not an idle pursuit. In 1972, Hugh Montgomery, a mathematician at the University of Michigan, investigated the statistical distribution of the zeros. He found that they were scattered randomly but seemed to repel each other slightly - they did not clump together. On a trip to the Institute for Advanced Study in Princeton he showed his result to the physicist Freeman Dyson. By sheer luck, Dr. Dyson was one of the few people in the world who would have recognized that the Montgomery results looked just like recent calculations on the energy levels of large atoms. The coincidence was so striking that it forged a new and still mysterious bridge between quantum physics and number theory. ... When ... Dr. Andrew Wiles of Princeton solved ... Fermat's conjecture ... in 1993, after working for seven years in secrecy ... his path ... had been ... clear: recent results had indicated the most promising direction to travel. Mathematicians at the conference agreed that there was no such clear evidence of a trail head for the Riemann hypothesis, a challenge they called both frustrating and exhilarating. "The Riemann hypothesis is not the last word about things," Dr. Montgomery said. "It should be the first fundamental theorem. We're in a kind of logjam right now because we can't prove the fundamental theorem." ...".

Carlos Castro and Jorge Mahecha

have written a paper hep-th/0208221 entitled Final Steps Towards a Proof of the Riemann Hypothesis. In January 2003, Carlos Castro said in an e-mail message: "... Jorge and I sent sometime ago to a journal the Hilbert-Polya proposal to prove the Riemann Hypothesis based on a Fractal Supersymmetric QM model. This is the way to prove it. ... The other proposal based on our paper on the web has a Godel-Chaitin element to it. Chaitin ( a computer scientist ) refined Godel's work to show that there are random truths which cannot be proven. In our paper, there are orbits in the R x C ( reals times complex ) defined by : Z [ A ( s - 1/2 ) + 1/2 ] = Z ( s' ) For ANY given s', one has an orbital of points ( A, s ) in R x C . Whether or not it is true that *outside* the Riemann critical line one has: Z [ 1/2 + i0 ] = Z ( s' ) for some *unknown* s', cannot be proven or disproven. Because the limit of < 1/2 + i0 | s > = Z [ A ( s - 1/2 ) + 1/2 ] when A goes to infinity, and s goes to 1/2 + i0 , IS ***path dependent***. It may, or may not, be equal to Z [ 1/2 + i0 ], when Z ( s' ) = real, for an *unknown* s' that is lying outside the critical Riemann line but inside the critical complex domain. This is the Chaitin-Godel theorem, there are random truths which cannot be proven. However, our proof based on the Hilbert-Polya propsal avoids this Chaitin-Godel element to it , and it works thanks to QM ! ...".

The Castro/Mahecha paper was submitted to Annals of Mathematics, and on 24 June 2003 Carlos Castro said, in an e-mail message:

"... We got a response from the Annals of Mathematics journal at Princeton related to my paper with Jorge on the RH. In one sentence, the referee said that it was not enough to find x = 1/2 as a fixed point. Since the y-values of the zeros move, we should find arguments to derive the location of the y-values as well. We have done so in our recent paper where we have a Fractal Supersymmetric QM model whose spectrum yields the y-values of the zeta zeros. ...".

On 30 June 2003 Jorge Macheda said, in an e-mail message about posting to the Cornell arXiv the paper that answered the question of the referee:

"... Yesterday I posted the paper, it had the number math-ph/0306077 ...".

Later that day, 30 June 2003, I checked the Cornell arXiv, and sent a message to Jorge, Carlos, et al, saying:

"... Jorge, Carlos, and everybody, I see that the paper at is NOT the RH paper. It is, instead, Structures of boson and fermion Fock spaces in the space of symmetric functions by Yurii A. Neretin. Therefore, I conclude that the blacklist against Carlos is still quite effective. ...".

Since Carlos Castro is blacklisted by Cornell from posting to arXiv, I have put on my web site the paper

A fractal SUSY-QM model and the Riemann hypothesis,

by Carlos Castro ... Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia, USA ... , Jorge Mahecha ... Institute of Physics, University of Antioquia, Medellin, Colombia ... June 29, 2003 ... .

The paper (in pdf format) is here. [A later 9 May 2004 pdf paper is here.]

Its abstract states:

"... The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s = 1/2 + i lamdba_n. Hilbert-Polya argued that if a Hermitian operator exists whose eigenvalues are the imaginary parts of the zeta zeros, lambda_n's, then the RH is true. In this paper a fractal supersymmetric quantum mechanical (SUSY-QM) model is proposed to prove the RH. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) that is present in the fractal analog of the CBC (Comtet, Bandrauk, Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter beta is one-half the fractal dimension of the Weierstrass function. An ordinary SUSY-QM oscillator is constructed whose eigenvalues are of the form lambda_n = n pi and which coincide with the imaginary parts of the zeros of the function sin(iz). This sine function obeys a trivial analog of the RH. A review of our earlier proof of the RH based on a SUSY QM model whose potential is related to the Gauss-Jacobi theta series is also included. The spectrum is given by s(1 - s) which is real in the critical line (location of the nontrivial zeros) and in the real axis (location of the trivial zeros). ...".

In an attempt to justify their blacklisting, Cornell (by a 19 February 2003 e-mail message from " (moderation for" (whatever or whoever that is) contended that Carlos Castro was not affiliated with Clark Atlanta University (Cornell misspelled "Atlanta" as "Atlantic".), whereupon on 1 July 2003 Carlos Handy sent a e-mail to Cornell at "moderation for" <> saying

"... To Whom It May Concern:
My name is Dr. Carlos R. Handy, Co-Director of the Center for
Theoretical Studies of Physical Systems, and well known to Dr.
Fred Cooper, formerly at LANL T-8 (where I served as post-doc,
1978 - 81), and now at NSF.
Dr. Carlos Castro, an American citizen, is an affiliate of our Center,
and has been one for at least six years.  He has published papers
with the name of our center, and university (Clark Atlanta University,
not ``Atlantic''). We recognize that he is controversial, but in those
matters of particular concern and interest to us, he has always been
very insightful and credible. We do not like the impression of an
intellectual mafia that seems to be engaged in scientific censorship.
More specifically, in the past, Dr. Ginzparg has made e-mail remarks
to Dr. Fred Cooper, concerning our support of Dr. Castro, that we have
regarded as prejudiced  and offensive. This should be particularly troubling to
your organization, which continues to enjoy government support.
I would be glad to submit a written letter, but I trust, with the particular
reference to Dr. Cooper, that this might not be necessary. In any case,
please send me your specific address, and I will mail such a letter
I understand that your archives should not be the repository for any
irresponsbile, purportedly scientific articles, however, we (CTSPS)
do not support everyone, and until such time that Dr. Castro behaves
in a manner questionable to us, we will continue to support him.
Dr. Carlos R. Handy
Associate Professor of Physics
Co-Director, CTSPS ...". 
Despite the affirmation that Carlos Castro is indeed affiliated with Clark Atlanta University, Cornell has, as far as I know, continued to blacklist him.


Primes as Simple Representations

As Heinrich Saller has remarked: 
Boole said that 
mathematics is not primarily about numbers, 
but structures.
For the integers Z, 
the primes define the maximal principal ideals pZ, 
the tips of the number tree, 
partially ordered by divisibility as order relation. 
One has a model for representations -
primes correspond to simple representations.
Products of different primes 
correspond to semisimple representations. 
Products of one prime 3, 9, 27, 81, ... are monogeneous and 
correspond to representations with nilpotent contributions.
The decomposition of a representation 
into nondecomposable ones 
is the unique prime powers decomposition. 
(e-mail from Heinrich Saller 24 Jan 97)

According to a Prime Curios! web page:

"... What folks often forget is a program (any file actually) is a string of bits (binary digits)--so every program is a number. Some of these are prime.

Phil Carmody found this one

8565078965 7397829309 8418946942 8613770744 2087351357 
9240196520 7366869851 3401047237 4469687974 3992611751 
0973777701 0274475280 4905883138 4037549709 9879096539 
5522701171 2157025974 6669932402 2683459661 9606034851 
7424977358 4685188556 7457025712 5474999648 2194184655 
7100841190 8625971694 7970799152 0048667099 7592359606 
1320725973 7979936188 6063169144 7358830024 5336972781 
8139147979 5551339994 9394882899 8469178361 0018259789 
0103160196 1835034344 8956870538 4520853804 5842415654 
8248893338 0474758711 2833959896 8522325446 0840897111 
9771276941 2079586244 0547161321 0050064598 2017696177 
1809478113 6220027234 4827224932 3259547234 6880029277 
7649790614 8129840428 3457201463 4896854716 9082354737 
8356619721 8622496943 1622716663 9390554302 4156473292 
4855248991 2257394665 4862714048 2117138124 3882177176 
0298412552 4464744505 5834628144 8833563190 2725319590 
4392838737 6407391689 1257924055 0156208897 8716337599 
9107887084 9081590975 4801928576 8451988596 3053238234 
9055809203 2999603234 4711407760 1984716353 1161713078 
5760848622 3637028357 0104961259 5681846785 9653331007 
7017991614 6744725492 7283348691 6000647585 9174627812 
1269007351 8309241530 1063028932 9566584366 2000800476 
7789679843 8209079761 9859493646 3093805863 3672146969 
5975027968 7712057249 9666698056 1453382074 1203159337 
7030994915 2746918356 5937621022 2006812679 8273445760 
9380203044 7912277498 0917955938 3871210005 8876668925 
8448700470 7725524970 6044465212 7130404321 1826101035 
9118647666 2963858495 0874484973 7347686142 0880529443 

in March 2001.

When written in base 16 (hexidecimal), this prime forms a gzip file of the original C-source code (sans tables) that decrypts the DVD Movie encryption scheme (DeCSS). See Gallery of CSS Descramblers (and its Steganography Wing) for more information.

It is apparantly illegal to distribute this source code in the United States, so does that make this number (found by Phil Carmody) also illegal? ...".


References, Acknowledgements, etc:

Thanks to Liam Roche at for correcting some of my mistakes about prime numbers.



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