June 2014 - mathfun@mailman.xmission.com

[math-fun] The first congruent number
by Bill Gosper 18 Jul '14

18 Jul '14

In what was probably Europe's first public math contest, the great Fibonacci was asked for three rational squares with common difference 5. (This is equivalent to finding a rational Pythagorean triangle with area 5.) Somehow, Fibonacci found (31/12)^2, (41/12)^2, (49/12)^2. This is the first of an infinite sequence of such solutions, found these days by repeated "addition" of the point {5*5/4, 6*5^2/4^2} along the "elliptic" curve y^2 = (x-5) x (x+5), producing the sequence 5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, ..., whose elements *of even index* are Fibonacci's desire. Elliptic curve addition doesn't quite provide a nice recurrence formula. Empirically, there's a mysterious, period 8 sign pattern: pisano[1] = 5/2; pisano[2] = 41/12; pisano[n_Integer] := (5 \[Sqrt](-100 + 9 pisano[-1 + n]^2 + 4 pisano[-1 + n]^4 - 12 (-1)^(1/8 (-2 + n) (1 + n) (3 + n) (4 + n)) Sqrt[-25 pisano[-1 + n]^2 + pisano[-1 + n]^6]))/ (Abs[-5 + 2 pisano[-1 + n]] (5 + 2 pisano[-1 + n])) Try a few: Table[pisano[n], {n, 8}] {5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, 312098738002194296165/128615821825334210638, 249850594047271558364480641/5354229862821602092291248} Test the "even" ones: Sqrt[#^2 + {-5, 5}] & /@ %[[Range[2, 8, 2]]] {{31/12, 49/12}, {113279/1494696, 4728001/1494696}, {518493692732129/178761481355556, 767067390499249/178761481355556}, {249563579992463717493803519/5354229862821602092291248, 250137278774864229623059201/5354229862821602092291248}} Rich suggested the solutions might obey a Somos recurrence. So far, I've only found one (Somos5) for the numerators: a[n] -> (124558 a[-3 + n] a[-2 + n] - 781 a[-4 + n] a[-1 + n])/a[-5 + n] The 124558 and 781 are very probably minimal. "Fitting" a 𝝑 fcn to these numerators held a surprise: a[0], a[1], a[2], ... , a[n] = 1,5,41,11285,... = (I (-1)^n 5^(1/2 Mod[n, 2]) E^(- I ArcTan[3/5]/2) (π/2)^(3/4) u^n^2* EllipticTheta[2, n z, (-1)^(-(13/34) + I/34)])/(2 17^(1/4) Gamma[5/4]) (Note the crazy (constructible!) 34th root of unity. Unfortunately, u and z are simply what they need to be to make a[1] and a[2] work. ISC doesn't recognize them, and I can't find a way to symbolically separate them. z -> 4.908105290682877137963508159098016296309287655031188041917052851155165374957836660235707132115102`69. - 0.197404925101320072627868756915313864010305639948302119404764160895568125915396715987289924743617`69. I, u -> -0.910971694653082823579468669395203737157839231764166792826687755862129892782777001356185727331806`65.8+ 1.430029427800740885258427207242669226859970344275381501850156729643487583694210398313066894949425`66. I} Mathematica's Root notation lets you exactly specify an implicitly defined root, say, the solution to a transcendental equation. Newton's method is getting to be like another button on your calculator. But how do we handle the solution vectors of simultaneous transcendental (say) equations, like for u and z here? --rwg (with much help from NeilB)

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[math-fun] Quantum Mechanical Pilot waves: they're baaaack !
by Henry Baker 01 Jul '14

01 Jul '14

FYI -- Follow the link to also see the pix. http://www.wired.com/2014/06/the-new-quantum-reality/ Have We Been Interpreting Quantum Mechanics Wrong This Whole Time? By Natalie Wolchover, Quanta Magazine 06.30.14 6:30 am For nearly a century, ÂrealityÂ has been a murky concept. The laws of quantum physics seem to suggest that particles spend much of their time in a ghostly state, lacking even basic properties such as a definite location and instead existing everywhere and nowhere at once. Only when a particle is measured does it suddenly materialize, appearing to pick its position as if by a roll of the dice. This idea that nature is inherently probabilistic Â that particles have no hard properties, only likelihoods, until they are observed Â is directly implied by the standard equations of quantum mechanics. But now a set of surprising experiments with fluids has revived old skepticism about that worldview. The bizarre results are fueling interest in an almost forgotten version of quantum mechanics, one that never gave up the idea of a single, concrete reality. The experiments involve an oil droplet that bounces along the surface of a liquid. The droplet gently sloshes the liquid with every bounce. At the same time, ripples from past bounces affect its course. The dropletÂs interaction with its own ripples, which form whatÂs known as a pilot wave, causes it to exhibit behaviors previously thought to be peculiar to elementary particles Â including behaviors seen as evidence that these particles are spread through space like waves, without any specific location, until they are measured. Particles at the quantum scale seem to do things that human-scale objects do not do. They can tunnel through barriers, spontaneously arise or annihilate, and occupy discrete energy levels. This new body of research reveals that oil droplets, when guided by pilot waves, also exhibit these quantum-like features. To some researchers, the experiments suggest that quantum objects are as definite as droplets, and that they too are guided by pilot waves Â in this case, fluid-like undulations in space and time. These arguments have injected new life into a deterministic (as opposed to probabilistic) theory of the microscopic world first proposed, and rejected, at the birth of quantum mechanics. ÂThis is a classical system that exhibits behavior that people previously thought was exclusive to the quantum realm, and we can say why,Â said John Bush, a professor of applied mathematics at the Massachusetts Institute of Technology who has led several recent bouncing-droplet experiments. ÂThe more things we understand and can provide a physical rationale for, the more difficult it will be to defend the Âquantum mechanics is magicÂ perspective.Â Magical Measurements The orthodox view of quantum mechanics, known as the ÂCopenhagen interpretationÂ after the home city of Danish physicist Niels Bohr, one of its architects, holds that particles play out all possible realities simultaneously. Each particle is represented by a Âprobability waveÂ weighting these various possibilities, and the wave collapses to a definite state only when the particle is measured. The equations of quantum mechanics do not address how a particleÂs properties solidify at the moment of measurement, or how, at such moments, reality picks which form to take. But the calculations work. As Seth Lloyd, a quantum physicist at MIT, put it, ÂQuantum mechanics is just counterintuitive and we just have to suck it up.Â A classic experiment in quantum mechanics that seems to demonstrate the probabilistic nature of reality involves a beam of particles (such as electrons) propelled one by one toward a pair of slits in a screen. When no one keeps track of each electronÂs trajectory, it seems to pass through both slits simultaneously. In time, the electron beam creates a wavelike interference pattern of bright and dark stripes on the other side of the screen. But when a detector is placed in front of one of the slits, its measurement causes the particles to lose their wavelike omnipresence, collapse into definite states, and travel through one slit or the other. The interference pattern vanishes. The great 20th-century physicist Richard Feynman said that this double-slit experiment Âhas in it the heart of quantum mechanics,Â and Âis impossible, absolutely impossible, to explain in any classical way.Â Some physicists now disagree. ÂQuantum mechanics is very successful; nobodyÂs claiming that itÂs wrong,Â said Paul Milewski, a professor of mathematics at the University of Bath in England who has devised computer models of bouncing-droplet dynamics. ÂWhat we believe is that there may be, in fact, some more fundamental reason why [quantum mechanics] looks the way it does.Â Riding Waves The idea that pilot waves might explain the peculiarities of particles dates back to the early days of quantum mechanics. The French physicist Louis de Broglie presented the earliest version of pilot-wave theory at the 1927 Solvay Conference in Brussels, a famous gathering of the founders of the field. As de Broglie explained that day to Bohr, Albert Einstein, Erwin SchrÃ¶dinger, Werner Heisenberg and two dozen other celebrated physicists, pilot-wave theory made all the same predictions as the probabilistic formulation of quantum mechanics (which wouldnÂt be referred to as the ÂCopenhagenÂ interpretation until the 1950s), but without the ghostliness or mysterious collapse. The probabilistic version, championed by Bohr, involves a single equation that represents likely and unlikely locations of particles as peaks and troughs of a wave. Bohr interpreted this probability-wave equation as a complete definition of the particle. But de Broglie urged his colleagues to use two equations: one describing a real, physical wave, and another tying the trajectory of an actual, concrete particle to the variables in that wave equation, as if the particle interacts with and is propelled by the wave rather than being defined by it. For example, consider the double-slit experiment. In de BroglieÂs pilot-wave picture, each electron passes through just one of the two slits, but is influenced by a pilot wave that splits and travels through both slits. Like flotsam in a current, the particle is drawn to the places where the two wavefronts cooperate, and does not go where they cancel out. De Broglie could not predict the exact place where an individual particle would end up Â just like BohrÂs version of events, pilot-wave theory predicts only the statistical distribution of outcomes, or the bright and dark stripes Â but the two men interpreted this shortcoming differently. Bohr claimed that particles donÂt have definite trajectories; de Broglie argued that they do, but that we canÂt measure each particleÂs initial position well enough to deduce its exact path. In principle, however, the pilot-wave theory is deterministic: The future evolves dynamically from the past, so that, if the exact state of all the particles in the universe were known at a given instant, their states at all future times could be calculated. At the Solvay conference, Einstein objected to a probabilistic universe, quipping, ÂGod does not play dice,Â but he seemed ambivalent about de BroglieÂs alternative. Bohr told Einstein to Âstop telling God what to do,Â and (for reasons that remain in dispute) he won the day. By 1932, when the Hungarian-American mathematician John von Neumann claimed to have proven that the probabilistic wave equation in quantum mechanics could have no Âhidden variablesÂ (that is, missing components, such as de BroglieÂs particle with its well-defined trajectory), pilot-wave theory was so poorly regarded that most physicists believed von NeumannÂs proof without even reading a translation. More than 30 years would pass before von NeumannÂs proof was shown to be false, but by then the damage was done. The physicist David Bohm resurrected pilot-wave theory in a modified form in 1952, with EinsteinÂs encouragement, and made clear that it did work, but it never caught on. (The theory is also known as de Broglie-Bohm theory, or Bohmian mechanics.) Later, the Northern Irish physicist John Stewart Bell went on to prove a seminal theorem that many physicists today misinterpret as rendering hidden variables impossible. But Bell supported pilot-wave theory. He was the one who pointed out the flaws in von NeumannÂs original proof. And in 1986 he wrote that pilot-wave theory Âseems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.Â The neglect continues. A century down the line, the standard, probabilistic formulation of quantum mechanics has been combined with EinsteinÂs theory of special relativity and developed into the Standard Model, an elaborate and precise description of most of the particles and forces in the universe. Acclimating to the weirdness of quantum mechanics has become a physicistsÂ rite of passage. The old, deterministic alternative is not mentioned in most textbooks; most people in the field havenÂt heard of it. Sheldon Goldstein, a professor of mathematics, physics and philosophy at Rutgers University and a supporter of pilot-wave theory, blames the ÂpreposterousÂ neglect of the theory on Âdecades of indoctrination.Â At this stage, Goldstein and several others noted, researchers risk their careers by questioning quantum orthodoxy. A Quantum Drop Now at last, pilot-wave theory may be experiencing a minor comeback Â at least, among fluid dynamicists. ÂI wish that the people who were developing quantum mechanics at the beginning of last century had access to these experiments,Â Milewski said. ÂBecause then the whole history of quantum mechanics might be different.Â The experiments began a decade ago, when Yves Couder and colleagues at Paris Diderot University discovered that vibrating a silicon oil bath up and down at a particular frequency can induce a droplet to bounce along the surface. The dropletÂs path, they found, was guided by the slanted contours of the liquidÂs surface generated from the dropletÂs own bounces Â a mutual particle-wave interaction analogous to de BroglieÂs pilot-wave concept. In a groundbreaking experiment, the Paris researchers used the droplet setup to demonstrate single- and double-slit interference. They discovered that when a droplet bounces toward a pair of openings in a damlike barrier, it passes through only one slit or the other, while the pilot wave passes through both. Repeated trials show that the overlapping wavefronts of the pilot wave steer the droplets to certain places and never to locations in between Â an apparent replication of the interference pattern in the quantum double-slit experiment that Feynman described as Âimpossible Â to explain in any classical way.Â And just as measuring the trajectories of particles seems to ÂcollapseÂ their simultaneous realities, disturbing the pilot wave in the bouncing-droplet experiment destroys the interference pattern. Droplets can also seem to ÂtunnelÂ through barriers, orbit each other in stable Âbound states,Â and exhibit properties analogous to quantum spin and electromagnetic attraction. When confined to circular areas called corrals, they form concentric rings analogous to the standing waves generated by electrons in quantum corrals. They even annihilate with subsurface bubbles, an effect reminiscent of the mutual destruction of matter and antimatter particles. In each test, the droplet wends a chaotic path that, over time, builds up the same statistical distribution in the fluid system as that expected of particles at the quantum scale. But rather than resulting from indefiniteness or a lack of reality, these quantum-like effects are driven, according to the researchers, by Âpath memory.Â Every bounce of the droplet leaves a mark in the form of ripples, and these ripples chaotically but deterministically influence the dropletÂs future bounces and lead to quantum-like statistical outcomes. The more path memory a given fluid exhibits Â that is, the less its ripples dissipate Â the crisper and more quantum-like the statistics become. ÂMemory generates chaos, which we need to get the right probabilities,Â Couder explained. ÂWe see path memory clearly in our system. It doesnÂt necessarily mean it exists in quantum objects, it just suggests it would be possible.Â The quantum statistics are apparent even when the droplets are subjected to external forces. In one recent test, Couder and his colleagues placed a magnet at the center of their oil bath and observed a magnetic ferrofluid droplet. Like an electron occupying fixed energy levels around a nucleus, the bouncing droplet adopted a discrete set of stable orbits around the magnet, each characterized by a set energy level and angular momentum. The ÂquantizationÂ of these properties into discrete packets is usually understood as a defining feature of the quantum realm. If space and time behave like a superfluid, or a fluid that experiences no dissipation at all, then path memory could conceivably give rise to the strange quantum phenomenon of entanglement Â what Einstein referred to as Âspooky action at a distance.Â When two particles become entangled, a measurement of the state of one instantly affects that of the other. The entanglement holds even if the two particles are light-years apart. In standard quantum mechanics, the effect is rationalized as the instantaneous collapse of the particlesÂ joint probability wave. But in the pilot-wave version of events, an interaction between two particles in a superfluid universe sets them on paths that stay correlated forever because the interaction permanently affects the contours of the superfluid. ÂAs the particles move along, they feel the wave field generated by them in the past and all other particles in the past,Â Bush explained. In other words, the ubiquity of the pilot wave Âprovides a mechanism for accounting for these nonlocal correlations.Â Yet an experimental test of droplet entanglement remains a distant goal. Subatomic Realities Many of the fluid dynamicists involved in or familiar with the new research have become convinced that there is a classical, fluid explanation of quantum mechanics. ÂI think itÂs all too much of a coincidence,Â said Bush, who led a June workshop on the topic in Rio de Janeiro and is writing a review paper on the experiments for the Annual Review of Fluid Mechanics. Quantum physicists tend to consider the findings less significant. After all, the fluid research does not provide direct evidence that pilot waves propel particles at the quantum scale. And a surprising analogy between electrons and oil droplets does not yield new and better calculations. ÂPersonally, I think it has little to do with quantum mechanics,Â said Gerard Ât Hooft, a Nobel Prize-winning particle physicist at Utrecht University in the Netherlands. He believes quantum theory is incomplete but dislikes pilot-wave theory. Many working quantum physicists question the value of rebuilding their highly successful Standard Model from scratch. ÂI think the experiments are very clever and mind-expanding,Â said Frank Wilczek, a professor of physics at MIT and a Nobel laureate, Âbut they take you only a few steps along what would have to be a very long road, going from a hypothetical classical underlying theory to the successful use of quantum mechanics as we know it.Â ÂThis really is a very striking and visible manifestation of the pilot-wave phenomenon,Â Lloyd said. ÂItÂs mind-blowing Â but itÂs not going to replace actual quantum mechanics anytime soon.Â In its current, immature state, the pilot-wave formulation of quantum mechanics only describes simple interactions between matter and electromagnetic fields, according toDavid Wallace, a philosopher of physics at the University of Oxford in England, and cannot even capture the physics of an ordinary light bulb. ÂIt is not by itself capable of representing very much physics,Â Wallace said. ÂIn my own view, this is the most severe problem for the theory, though, to be fair, it remains an active research area.Â Pilot-wave theory has the reputation of being more cumbersome than standard quantum mechanics. Some researchers said that the theory has trouble dealing with identical particles, and that it becomes unwieldy when describing multiparticle interactions. They also claimed that it combines less elegantly with special relativity. But other specialists in quantum mechanics disagreed or said the approach is simply under-researched. It may just be a matter of effort to recast the predictions of quantum mechanics in the pilot-wave language, said Anthony Leggett, a professor of physics at the University of Illinois, Urbana-Champaign, and a Nobel laureate. ÂWhether one thinks this is worth a lot of time and effort is a matter of personal taste,Â he added. ÂPersonally, I donÂt.Â On the other hand, as Bohm argued in his 1952 paper, an alternative formulation of quantum mechanics might make the same predictions as the standard version at the quantum scale, but differ when it comes to smaller scales of nature. In the search for a unified theory of physics at all scales, Âwe could easily be kept on the wrong track for a long time by restricting ourselves to the usual interpretation of quantum theory,Â Bohm wrote. Some enthusiasts think the fluid approach could indeed be the key to resolving the long-standing conflict between quantum mechanics and EinsteinÂs theory of gravity, which clash at infinitesimal scales. ÂThe possibility exists that we can look for a unified theory of the Standard Model and gravity in terms of an underlying, superfluid substrate of reality,Â said Ross Anderson, a computer scientist and mathematician at the University of Cambridge in England, and the co-author of a recent paper on the fluid-quantum analogy. In the future, Anderson and his collaborators plan to study the behavior of ÂrotonsÂ (particle-like excitations) in superfluid helium as an even closer analog of this possible Âsuperfluid model of reality.Â But at present, these connections with quantum gravity are speculative, and for young researchers, risky ideas. Bush, Couder and the other fluid dynamicists hope that their demonstrations of a growing number of quantum-like phenomena will make a deterministic, fluid picture of quantum mechanics increasingly convincing. ÂWith physicists itÂs such a controversial thing, and people are pretty noncommittal at this stage,Â Bush said. ÂWeÂre just forging ahead, and time will tell. The truth wins out in the end.Â

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Re: [math-fun] Robert Smith's vector problem...
by Henry Baker 01 Jul '14

01 Jul '14

Sorry, Robert, but I still don't understand. What is the meaning of "(v . <i, j, k>)" ? Do you simply mean a vector (= pure (imaginary) quaternion; i.e., S(v)=0) ? At 01:50 PM 6/26/2014, Robert Smith wrote: >Most formally speaking, for a quaternion Q and a vector v in R^3, and >for * meaning quaternion multiplication, > >Q # v = Q * (v . <i, j, k>) * Q^{-1} > >where i, j, and k are the pure unit quaternions. Originally I said >conj(Q) instead of Q^{-1} which is only correct some of the time. > >Robert > >On Thu, Jun 26, 2014 at 11:23 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I didn't understand your "#" operation. Could you please try again?

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[math-fun] Siimple probability problem
by Bernie Cosell 01 Jul '14

01 Jul '14

It is now clear that I've lost essentially all of my math skills... sigh. My wife asked what seemed to me to be a simple problem but I couldn't see a way to approach it... I kept going down what felt like blind alleys that just got more complicated rather than leading toward a solution: You have a jar with N *each* of C colors of marbles [that is, you start with the same number of each color, N*C marbles in the jar]. You start removing marbles [without replacement]. What is the expected number of marbles you have to remove to have a 50% chance of having at least one of each color. Is there some easy/obvious way to approach/analyze this? THANKS! /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] pilot waves in QM
by Warren D Smith 30 Jun '14

30 Jun '14

I'd thought they'd never managed to implement that idea in relativistic QM (meaning quantum field theory) so it was effectively dead. In particular consider the Conway-Kochen-Specker impossibility theorem about spin.

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[math-fun] some conjectures (theorems?) about quadratic forms
by Warren D Smith 30 Jun '14

30 Jun '14

Actually, the problem I'd really like to solve -- the one on positive definite quadratic forms was a special case of it -- is this. Instead of "positive definite quadratic forms in n variables" consider "polynomials in n variables which cannot produce negative output variables," and which input integers and output integers. Prove or disprove: there exists such a polynomial which assumes every natural number value (albeit perhaps we should permit a finite number of exceptions which cannot be represented) as output, and which represents each natural in at most a bounded number of ways. My theorem was a disproof in the case where the polynomial is quadratic. I suspect it can be disproven more generally, but if so, that is going to be harder... Pfister's solution of Hilbert's 17th problem is relevant to this.

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[math-fun] Jeff D. Caldwell quadratic form confusion by me
by Warren D Smith 30 Jun '14

30 Jun '14

I apologize to Jeff D. Caldwell, I apparently transposed 2 of his coefficients 1, 0,-1 instead of 1-,1,0. Anyhow, then ok. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] some conjectures (theorems?) about quadratic forms
by Warren D Smith 30 Jun '14

30 Jun '14

OK, here is a proof of the 2nd conjecture that I claimed before (but I should have said "positive definite" quadratic forms) THEOREM: Let Q be a positive-definite integer-valued quadratic form in some number of integer variables. Then either Q fails to represent an infinite subset of natural numbers, or Q represents each member of an (other) infinite subset of them in an unboundedly large number of ways, indeed growing like a power law. PROOF (OF THIS PLUS SOME MORE): Via the Hardy-Littlewood "singular series" one can prove (i.e. others have proven) that the number of representations of N as a sum of s terms, each a kth power of a natural number, behaves for N large like PositiveConstant(s,k)*N^(s/k +- o(1)) provided s is large enough compared to k. (And precise criteria are available for how large s must be.) But it is simpler if we do not ask for #representations = something+-small error, but instead merely ask that #reps >= PositiveConstant(s,k)*N^(s/k) and merely for an infinite subset of (but not all) N>0. Then this is trivially true by simply counting the number of ways you can choose s integers in the set [1/2, 1]*N^(1/k). There are order N^(s/k) ways, and they all produce reps of numbers in the set [1/2^k, 1]*s*N. Hence by pigeonhole principle some number in this set must be represented at least (1/s)*N^(s/k - 1) times. This count goes to infinity (and like a power law) when N-->infinity, provided only that s>k, In particular, any ternary (or quaternary,... or s-variate for any s>=3) integral positive definite quadratic form represents at least some integers (forming an infinite subset of natural numbers) in unboundedly many ways; the proof is similar by now counting lattice points in a ball of radius squareroot(N), and using pigeonhole principle. (Using the s-dimensional lattice corresponding to whatever s-variate quadratic form we are speaking of.) And "unboundedly many" is, more precisely, a power law. Now, what about quadratic forms which are not necessarily positive definite, or might be binary (2-variable) or only unary (1-variable)? Clearly a unary quadratic form Q(x)=a*x^2 cannot represent all integers, albeit those it does represent, are only represented in 2 ways. The binary *indefinite* form Q(x,y)=x*y represents each nonzero integer N in 4*#divisors(N) ways, which is always >=4 (and equals 4 if N=prime), and which is unboundedly large for highly composite N, but note the #divisors(N) function is upper bounded by N^o(1), i.e. grows slower than any positive power of N. [Jeff D. Caldwell also was suggested to me, erroneously, that you could say the same about the form x^2-y^2 = (x-y)*(x+y). That does not quite work because x-y and x+y necessarily have the same parity, hence this form cannot represent integers congruent to 2 mod 4. But his idea does work if applied to x*y.] A binary form Q(x,y)=a*x*x+c*y*y+b*x*y is positive definite if b*b < 4*a*c. I claim that every positive definite binary integral quadratic form (a) necessarily leaves an infinite subset of natural numbers unrepresented, and (b) necessarily represents some integers an unboundedly large number of ways. To prove (a), which was already known to C.F.Gauss and P.Fermat, let the "discriminant" of the form be d=4*a*c-b*b<0 and note that integers N cannot be represented if d is nonsquare modulo 4N. By quadratic reciprocity, this is an infinite positive density subset of N. Also (a) apparently was addressed in this paper: Valentin Blomer, Andrew Granville: Estimates for representation numbers of quadratic forms, Duke Math'l. J. 135,2 (2006) 261-302. which I think claims to prove that for any positive definite integer-valued binary quadratic form, the number of represented naturals below N is at least asymptotic to PositiveConstant * N / log(N)^(1/2). [But I have not actually read this paper -- can anybody provide PDF?] To prove (b) --which I provide as a bonus, since I do not actually need it -- consider the lattice of the complex numbers whose squared-norms are the quadratic form... this is a number ring... and my point is that that you can multiply complex numbers in the ring to get a norm which is represented in many ways. For example as the simplest case so you can see what I am talking about, Q(x,y) = x*x+y*y, the lattice is the just the Gaussian integers z=a+i*b with a,b ordinary integers, and then Q(x,y)=|z|^2. Now any norm is represented in 8 ways via a+ib, a-ib, b+ia, b-ia, -a-ib, -a+ib, -b+ia, -b-ia (well, only 4 if a=b, but ignore that as non-generic) but if we multiply (a+ib) * (c+id) then we get a norm generically represented in 16 ways, and (a+ib)*(c+id)*(e+if) will have norm generically represented in 32 ways, etc. My point is, in this manner you can construct numbers which have unboundedly many representations as sums of two squares; they arise from highly-composite Gaussian integers. Now similarly, if we work over any other complex number ring in place of "the Gaussian integers" we can do the same kind of construction -- highly composite ring members will yield natural numbers with unboundedly many representations by whatever positive definite binary quadratic form corresponds to the norm in that ring. This proves (b), indeed this construction should prove the maximum number (among natural numbers n<N) of representations of n must grow at least as quickly as 2^([1/2 - o(1)] * lnN / lnlnN) when N-->infinity. Q.E.D. EXTENSION: Theorem also valid for rational-valued quadratic forms with integer arguments. (Because just multiply by some constant denominator to get an integer valued form...) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The first congruent number (Bill Gosper)
by Warren D Smith 30 Jun '14

30 Jun '14

--seems like fantastically beautiful stuff.

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[math-fun] H.Baker's neutron star like "magic materials"
by Warren D Smith 29 Jun '14

29 Jun '14

>From: Henry Baker <hbaker1(a)pipeline.com> >"nuclear" materials made from the basic >stuff of neutrons & protons... >we used to think that diamonds >could only be made under extreme pressure, but >this has been found to be false; diamond >coatings can be made with with chemical vapor >deposition (CVD) under relatively low pressure... >What I'm thinking of is a kind of "foam" which is >a more-or-less fractal material made out of nuclear >"soup" (protons-cum-neutrons). Think Sierpinski >pyramids or Apollonian foams. Since the material >is mostly empty space, it could be considerably >less dense than the stuff of neutron stars. >...could it exist? --First of all, it could not have net positive charge (some percent protons) and still be a "bulk material." So you'd need electrons in the soup too, or it'd have to be neutrons only. (And you might as well just drop the fig leaf and just say it is made of of quarks.) Now actually a "hot soup" made of neutrons, protons, electrons, positrons, neutrinos, and photons at MeV to GeV temperatures presumably did exist in the early universe, and things would have stayed that way forever if there had not been expansion. But I assume Baker is not satisfied with that stuff, he wants his soup cold. Now if it were just neutrons only, then everybody believes it'd be unstable to beta decay, meaning you lose. Can this instability be proven, or is it merely a good guess? Probably the latter. I don't think anybody currently can *prove* Baker's magic material (should I presume we should call it "dough"?) cannot exist. If it were neutrons, protons, and electrons all mixed, then the protons could stabilize nearby neutrons via their attractive energy, and the electrons could provide overall charge neutrality. However that only works if enough protons are enough nearby each neutron (within about 1 fm). And if you try to make some sort of stringlike or treelike or frothlike filamentary structure with neutrons and protons along the strands, then you probably are going to be unstable due to enormous surface energy (nuclear matter has high surface tension). So... this is not a proof, and probably no proof can be produced with present day mathematics... but... I feel there is good reason to believe "Baker loses" i.e. no such magic material can exist. What however might exist is a so-called "island of stability" consisting of certain nuclei like flerovium-298, unbinilium-304 and unbihexium-310. See http://en.wikipedia.org/wiki/Island_of_stability This is based partly on theoretical understanding of nuclear energies from "shell models" and partly on voodoo, but it might be that certain transuranic elements with the right magic numbers of protons & neutrons might be able to exist with remarkably long half lives. In any case, nobody has been able to synthesize the highest-voodoo isotopes so nobody has been able to find out whether this speculation is true. There are some partial experimental indications that is it true, but the whole question looks out of reach of experiment, nobody has any feasible way to synthesize these beasts. You might think, though, that if it were true, then those beasts would have gotten made, and released into the outside universe, during neutron star collisions. Since nobody has detected these atoms, presumably they all are unstable with halflives always shorter than, say, 50 MYr (since otherwise we would have detected them).

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