Hidden Variables in the Bell Inequality Theorem? When non locality does not imply non causality   2 comments


SARS Coronavirus 2 update (March 27, 2020):


  • You will know that Newton, during the Great Plague that hit London and forced to close the Trinity Colle of Cambridge, took advantage of his confinement to develop his theory of gravity and  infinitesimal calculus that would determine the whole development of physics until the XX century. https://www.washingtonpost.com/history/2020/03/12/during-pandemic-isaac-newton-had-work-home-too-he-used-time-wisely/
  • Today we are experiencing a global pandemic, whose epicentre is now in Europe and the USA, that has confined us in our homes for an undetermined time. Could we expect that some people will take this difficult time to develop new and revolutionary theories that will change our view of Nature and our science and technology? We are seeing how some people with engineers’ minds are designing and developing or adapting in a in a very short period of time, using 3D technologies, devices that could supply the lack of respirators and ventilators in hospitals around the world; textil industries are adapting their production lines to make surgical masks in record times; and laboratories from half the world are developing faster and more accurate tests and already checking vaccines. In Spain we say that “need sharpens the wit”. And difficult and tough times and emergencies always bring out the best in us.
  • It has been quite surprising to see how every single country (with some exceptions in Asia, where they already have some experience about this kind of events), has made the same mistakes when it comes to taking actions against the Coronavirus pandemic expansion. All governments followed the advises of the Sanitary authorities, trusting on the expert scientists. And most of them – including the WHO – failed in their previsions in a scandalous way. When Italy was already suffering a exponential increasing of the cases, and having also the Chinese experience as a reference, experts kept saying that it was necessary to wait to the numeric data, the scientific proved fact that supposedly the number represents, to act in consequence. “Hypotheses non fingo” would have Newton said. Those that with simple common sense warned about what was obviously coming, were not listened to or were directly criticized. Some reputed experts – the same that would forme part of the advisory committees – even made jocks about the gravity of the new coronavirus pandemic.I’m afraid that the experts’ negligence was caused by their absolute confidence in their mathematical tools. Most scientists are not used to thinking rationally, they are only used to operating with algebraic formulas, making diferencial calculus and probabilistic approximations to try to understand and manipulate nature in their statistical reality. And when their mathematical models fail, they lose all their references and get totally paralysed  “Let’s wait to the next data to decide”, is the only thing they are able to babble. I have no doubt the forecast failures of so many experts has actually been a clamorous failure of our modern science, the science of mathematical calculators. It’s quite hilarious to see how people blindly trust in calculations as the only trustable scientific criterium. When they look for an authoritative argument, scientists say they have mathematical models that say… and when people listen to the words “algorithm”, “differential equations”, “statistical calculations” or “mathematical models” they feel that is something that must be secure, a scientific true.
  • The sanitary emergency and stress some countries are experiencing will be replaced at some point for an economic crisis whose consequences nobody actually knows yet but we can expect difficult times. People of different disciples are glimpsing that nothing will remain the same in our world.
  • I hope that, as in other truly emergency times of the past, we are going to experience a new scientific revolutions. We still have a pending and crucial revolution in physics and chemistry. We need such a revolution that lets us avoid the human experience fo scarcity drives us to the experience of the infinite material abundance that currently we are not able to create because we are not able to manipulate accurately the atomic nucleus. We need a new understanding of gravity, a new atomic model, and a new solar system model. But, friends, do not expect that revolution is going to come from our mainstream physicists, they are stuck, blind, and depressed since many years. The needed revolution is your own responsibility. Be sure that Newton and Einstein were not smarter than you.
    Be brave, be agile, be critic, trust in your reason, stay humble, and GO FOR IT.
  • —————————————————-

    When madness reigns, nothing is more revolutionary than reason.

    You will know that Einstein considered that quantum mechanics could not be a complete theory and that there must be some additional hidden variables. Those hidden variables should explain locally and causally the atomic reality. That would imply that if an action at one point has a consequence into another point there must exist a kind of physical space that connects those two points. So, that every effect must have a local and actual cause.

    See the EPR paradox here: https://en.wikipedia.org/wiki/EPR_paradox

    Here you can read the original Einstein – Podolsky article “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” https://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777

    But the EPR paradox was not accepted. If that distant effect happens at the same time in both separate points – as the entanglement principle states – the speed of light should not be limited by the special theory of relativity as it is. So, if the entanglement principle is right, quantum mechanics will be a non local theory. And it also implies the atomic realm is not ruled by causality.

    This is the conclusion shown by John S. Bell in his inequality theorem. If you know algebra you can read his article “On the Einstein, Podolsky, Rosen Paradox” where he mathematically proved that quantum mechanics could not be a local theory:


    The idea of non locality let physicists to develop quantum mechanics as a non causal but merely probabilistic model.

    We don’t need to know algebra though to realize that the notion of fundamental space assumed by Bell, although he speaks of subspaces, was not a dual composite space. He does not consider a dual intersecting system with shared subfields existing in and because of the intersection.

    Considering the next topological structure of two intersecting A and B fields that vibrate with the same or opposite phase, we don’t need to write algebraic equations to realize that if we change the phase of vibration of A (or of B), we are going to simultaneously affect, whatever the limit of the speed of light is, simultaneously to d, e, and f subfields. Non locality does not imply necessarily non causality.

    The problem of quantum mechanics is not about locality or non locality but about what idea of space we have assumed when we think about an atom and what we think entanglement means.

    Because the idea of space we have internally assumed is going to determine any other consideration we do about nature:

    The geocentric and heliocentric solar system models were built upon the assumption of a unique, independent, and invariant space orbited around its center. When Kepler realized that the planetary orbits were not circles, instead of thinking about a circular field that periodically expands and contracts, which mechanically could explain an elliptical trajectory and the acceleration and deceleration of a planetary velocity as the consequence of a pushing force or the lack of it, he focused his attention on the elipses as if they were real fields. But those elipses are actually imaginary figures we create when observing the planetary motions; the only real thing – at least in a first consideration – is the orbited field.

    Newton did not think about a varying space either when he developed his gravity as force of attraction, even though he was very aware such a magical force was not a mechanical explanation at all. The only possible way of explaining mechanically gravity is to consider it a force of pressure, as Newton’s friend Fatio and later Le Sage and many others did in different ways until the beginning of the XX century when such a hypothesis was finally forgotten because it implied the existence of an ether (and the ether was considered inexistent since the Michelson and Morley experiment). The Einstein’s gravitational curvature is not a mechanical explanation either because density and volume do not explain by them own the curvature of the space. [By density I don’t mean the ratio between weight and volume, but the spatial distribution inside of a volume which will determine its weight as a consequence of gravity. Gravity is what causes the weight, not weight what causes gravity, but gravity is caused and determined – in my opinion – by the resistance of a density. Einstein’s gravitational curvature does not explain what weight is. It would require to consider that weight is an ‘intrinsecal” property of matter, being that weight what curves the space, as some people think atomic electronegativity is; and that is not an enough explanation because all material properties are the consequence of something, not essential qualities that exist because that is its own nature of being].

    I think the curvature of a gravitational field only can be caused by a force of pressure, which implies something that resists such a pressure in some extent. If there’s a dense distribution of a space that is met by some kind of fluid in motion, such density will resist to be passed through. If the resistance is high, a diffracted curved field will be formed with the non passing fluid and the passing fluid will refract; that will imply a higher friction that will change the distribution of the resistant space, that will become less dense. And if there is less resistance, the diffraction of the curvature will be lower and the refraction of the passing fluid will be reduced.

    I’m thinking here about gravitational fields as longitudinal waves that that vibrate expanding and contracting periodically.

    The periodical variation of gravitational fields has not been measured, though. By the contrary, the periodical fluctuations and detected asymmetries of our solar system have been very well measured but physicists do not have yet any mechanical explanation of them. An no one seems to be very worried about those unexplained asymmetries. If Copernicus would have known that the planetary orbits are not circular but elliptical, that planetary velocities are not constant, that all the orbital elipses have a different eccentricity, that each planet has a different inclination, that even some planets rotate in an opposite direction, and that we do not have any mechanical and unique explanation of it – we only have remote hypothesis created ad hoc for each specific case and an abstract mathematical development – he undoubtedly would question the heliocentric model as well. Because Copernicus questioned the geocentric model just because it was totally asymmetric with no apparent reason.

    Anyway, a unique, independent but already vibrating field could explain gravity as a force of pressure but it cannot explain either the fluctuations nor the asymmetries of all the solar system planets. Maybe we are missing something yet. That is the what Einstein thought when it came to the atomic level and quantum mechanics.

    When it comes to the atom, it became soon clear that the model of a unique, independent and invariant field orbited around its center only could explain the behavior of the Hydrogen atom but anything else.

    Could explain mechanically the electromagnetic atom a unique wave pilot? Neither.

    In spite of spite Einstein’s opposition, I suppose time urgencies forced physicists to develop the atomic model in a utilitarian way, and as a pure abstract algebraic development without any visual reference, arriving to a merely probabilistic model when reason, logic and causal common sense, have nothing to say, barely following intuitions about symmetry some of them were able to blindly deduce from the equations. The atomic model was developed without reasonably understand anything, without any possibility of thinking rationally about it in a future, and without having any visual representation of the atom except a very diffuse and unique circle where statistical calculations are made on.

    (Today’s new attempts to developing a more consistent atomic model are so stuck, think about strings theories, without being able to get already clues from the equations, that some physicists are starting to consider if nature is not only irrational and random but also asymmetric without any reason and naturally deformed. You can deduce their level of desperation.)

    But what would it happen if we intersect or partially merge two longitudinal waves? In their intersection are going to be four vibrating subfields. Could be these subfields the subatomic particles that would form a nucleus shared by a dual composite atomic system? That would imply a change of paradigm. From a monist paradigm to a dualist paradigm. Many worlds interpretation already was proposed in the XX century, but multiverses have not been considered as the consequence of the intersection of parallel vibrating universes.

    Let’s describe this composite structure in a general way first, which is not any hypothesis but the representation of an actually composite topological structure:

    We start with two intersecting fields A (left) and B (right) that vibrate; in their intersection there will be 4 subfields – left and right, and up and down – shared by A and B. We can think about those 4 subfields as the “nucleus” shared by the dual composite system that is AB.

    1. If A and B vibrate with opposite phase, the left and right subfields will have as well opposite behaviors at a same moment: when A contracts and B expands, the left handed subfield will contract and the right subfield will expand, and viceversa; also there will be a subfield moving in a pendular way right to left or left to right towards the side of the contracted A or the contracted B respectively. So, we can know the exact situation and properties of A or B and of all their subfields only by measuring the situation of A or B at a specific moment.

    2. But if A and B vibrate with the same phase, some changes will take place in the behavior of the subfields. In this case, when A and B contract at the same moment, there will be two expanding mirror symmetric subfields at the left and right sides of the center of symmetry of the system; and a moment later, when A and B expand the left and right mirror symmetric subfields will contract. So here, the “contrary state” will not work (except when it comes to the contrary direction of their inner orbital motion). In this case there won’t be a left to right displacement of the other subfield but an upward (getting contracted) and downward (getting expanded) movement.

    In both scenarios, the AB’s subfields would have an inner orbital motion that would accelerate when the subfields contract and will decelerate when they expand; the direction of that inner motion will be different depending on where the subfield is placed.

    Now, let me suggest you how I would describe in my heterodox way the composite AB intersecting system if it were representing a composite dual atom with a unique shared nucleus:

    1. When the phases of variation of AB are opposite, the AB’s subfields would be fermions ruled by the Pauly Exclusion Principle. I think what you’re calling the “contrary state” is actually the consequence of the Pauly Exclusion Principle. To me the PEP does not mean that two subfields cannot exist at the same moment in the same place having the same properties, which is actually impossible in any case, but that two mirror symmetric subfields cannot exist at the left and right (or up and down) places with the same properties (expanded or contracted) at the same moment.

    In this case, when A is contracted and B expanded, the subfield moving towards left will be an electron causing a pushing pressure force towards left; a moment later, when A is expanded an B contracted, the electron subfield will move towards right becoming a positron. So electron and positron will be Majorana antiparticles as they both are the same subfield that is its own antimatter at different moments. When the electron subfields exists at the left side, we can say at the right side will “virtually” exist a positron, in the sense that the positron does not exist actually there yet but it potentially exists because it will become existent there a moment later.

    In this case as well, the contracted subfield existing at the left side when A is contracted, will be a contracted neutron; at that same moment, at the right side there will be an expanded anti-neutrino. A moment later, when A expands, the left contracted neutron will expand becoming a left expanded neutrino while the right expanded antineutrino will contract becoming a right contracted proton. In this sense, neutron and proton will be Dirac antiparticles because being different subfields they are mirror symmetric at different moments. The same can be told about the neutrino and antineutrino.

    I consider quarks in this context will be the pushing forces caused by the internal or external sides of the fields A and B when contracting or expanding. In this sense, electrons and positrons must be also be formed by quarks. Why should not, because no other reason that they have not been measured yet?

    2. When the phases of vibration of AB synchronize becoming equal, when AB contract, the electron-positron subfield will experience different forces that will cause its contraction and its upward motion following an orthogonal path at the centre of the symmetry of the system. The pushing force cause by this ascending subfield will be a photon. When AB expand, that upward contracted subfield will experience a decay of its inner orbital energy and it will lost its ascending pushing force. Here, the counter part of that lost force will appear at the convex side of the intersection of AB, creating an anti-photon with an inverted direction. If we, as observers, are placed at the concave side of the AB system, we won’t be able to detect the anti-photon when the AB expand and we could could think for us it’s a “dark” or invisible energy.

    At the left and right sides of the center of symmetry we will have two mirror symmetric subfields with the same expanded or contracted properties at the same moment, being the antimatter of each other. So they will be bosons as they are not ruled by the Pauly Exclusion Principle. The photonic subfield, although being a boson as well, will be ruled by the Pauly Exclusion Principle with respect to its antimatter as the photon and the anti-photon cannot exist at the same moment. (I’m not sure about the terminology here but maybe those left and right subfields could be electronic neutrino and antineutrinos when they are expanded).

    In both 1 and 2 cases, the subfields cannot be described with the spacial coordinates of A and B. (For example, the X coordinate of A or B will be the Y coordinate of the left and right subfields). So, those inner subfields will have extra espacial dimensions.

    Considering the fermionic and the bosonic systems as a unique topological system that gets periodically transformed into the other when the phases of vibration synchronize or desynchronize, it cannot be described with only a one time dimension. It will be necessary to consider two time fermionic dimensions that converge into one time bosonic dimension which diverge into two fermionic dimensions and so on.

    As a topologically transformative structure that evolves in a periodical loop way, the pushing forces that are the quarks (which are represented as flags or vectors in the diagrams) are supersymmetric. To describe supersymmetry – the symmetry between the fermions and bosons through time – in this context is not necessary to look for new subfields. Are the same subfields – whose shapes and material properties are transformed through time – which are supersymmetric:

    The next animation shows the fermionic and bosonic systems as separated figures to represent the different ways the quarks will change (the animation does not show the rotation of the systems around their center as it should).

    This composite model is actually a quantum machine. The quantum computer that Google or IBM can built is not atomic computer, it’s a machine that tries to emulate the quantum behavior following the probabilistic model that still is being accepted by the orthodoxy. Their “quantum supremacy” is a jock.

    It can be also represented in this way:

    With respect to the mathematics of this model, because today people do not consider descriptive geometry as true mathematics, you could find similitudes between the mentioned subfields and the Kaluza-Klein 5th dimension space which is simply a traversal cut in a string. The same transversal section is where Calabi Yau manifolds would be located.

    Take a look to the pictures provided at pages 12 and 15 in the book “The Shape of Inner Space” by Shing-Thu Yau:

    The Kaluza-Klein space is though a flat space while the subfields that appear when intersecting two fields are hyperbolic and being extradimensionals, are not one dimensional spaces.

    Actually, the few figures that Lobachevsky was able to draw – having a perfect algebra his developments about his imaginary geometry were not accepted by his mathematical contemporaries (While Gauss kept silent about the matter) – reminds as well the mentioned subfields:

    The intersecting spaces and subspaces can be as well considered as Riemann manifolds and submanifolds. Riemann already considered the spaces that appear when intersecting – that not overlapping – some spaces:

    The topological composite structure I presented – which gets different properties and shapes remaining the same surface – also is related to cobordism, and I think to cohomologies and link homologies. I asked several math professors as professors but or I didn’t get answer or they told they didn’t know it because I guess they are only used to seeing the algebraic abstract equations and they do not have a referential visual geometric representation of the structures they work with.

    I watched a video of a professor of Physics and mathematics, Jim Gates, speaking about supersymmetry. He developed a visual representation of the supersymmetry algebras and compact subalgebras related to internal symmetries that he and his colleagues called with the African term “Adinkra”: https://en.wikipedia.org/wiki/Adinkra_symbols_(physics)

    You can read more about Gates’ adinkras here https://web.archive.org/web/20160418235930/http://www.onbeing.org/blog/symbols-power-adinkras-and-nature-reality/2438

    It’s intersecting when he said “It can be shown that each adinkra corresponds to a distinct set of super-differential equations. Super-differential equations involve both the ordinary derivative operator (invented by Newton and Leibnitz) and a newer type of operator called a “super derivative”, which was invented in the mid-1970s by the mathematician Felix Berezin and then elaborated on by the physicists Abdus Salam and John Strathdee. Super derivatives, represented by the links in an adinkra, are similar to the ordinary derivative, except that they are allowed to violate the usual product rule for derivatives.”

    Of course i sent an email to professor Gates, let’s see if he has time to take a look to the above ideas. In other videos he speaks about a book he recently published: “Proving Einstein right”


    Also I recently sent these diagrams and explanations to professor Lee Smolin who has recently published a book about the Einstein incomplete revolution, linking them to a conference he gave at the Perimeter Institute, especially to the entanglement principle, to make him see that he had assumed a monistic idea of space that could be wrong. 1 hour later he kindly replied me “Thanks for your kind words and your email. I don’t really understand what your theory does, but the drawings are beautiful.” I think he cannot be an idiot, so he probably must be a very busy person and he only read the first and the last line and took a quickly look at the attached pictures.

    See what he says about entanglement in this conference and be aware how crazily lost these people are:

    Another different approach (because physicists and mathematicians do different approaches to the same thing with different tools and new invented names thinking they are discovering the Mediterranean every time, is to consider the dual composite system as a double quaternion system, or an octonion. The octonion approach has been developed in some extent by professor Cohl Furey (I also recently wrote her but she didn’t reply):


    Other approaches that can be considered (by always in an utilitarian way, without understanding the fundamental mechanism) are those who think in terms of knots (knots are intersections actually), for example yo can see what these chemistries are trying to do: https://www.quantamagazine.org/scientists-learn-the-ropes-on-tying-molecular-knots-20181029/

    The knot theory was already proposed by Lord Kelvin. See “The Vortex Theory of Atoms – pinnacle of classical physics”: https://t.co/ojavimtiXp

    And see https://en.wikipedia.org/wiki/Knot_theory

    But I think knots are a limited – schematic – way of representing the topological structures that atoms and the molecules that can be created with them are, it’s like to look at the system only from the top:

    Why do physicists and mathematicians not have ever consider the simple figure of two intersecting fields and their subfields is an interesting question. It seems they are quite obsessed and stuck with the geometric figure they call “Torus” which is a donut with a whole in its center. But a donut cannot explain the electromagnetic structure of fundamental matter in a mechanical and causalistic way.

    On the other hand, if you are interested on the subject of causal or ontological interpretations of quantum mechanics you could like to take a look to the the EmQM17 Symposiums:

    EmQM17 – Towards Ontology of Quantum Mechanics and the Conscious Agent


    They provide a free pdf version of the book “Emergent Quantum Mechanics David Bohm Centennial Perspectives”: https://www.mdpi.com/books/pdfview/book/1203

    And this is the researching lab of one of the co-coordinators of the symposium: https://phenoscience.com/video-media/

    Finally, I just saw this other book that seems to be interesting, I’d like to read it soon: “The Philosophy of Quantum Physics (By Cord Friebe, Meinard Kuhlmann, Holger Lyre, Paul M. Näger, Oliver Passon, Manfred Stöckler)


    Have a good day my friends.


    2 Respuestas a “Hidden Variables in the Bell Inequality Theorem? When non locality does not imply non causality

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    1. Estimado Alfonso, aunque no tengo el nivel de matemáticas para seguir tus aportaciones, me permito dejar aquí un enlace a una entrada que vi hace poco relativa al problema del “cubo perfecto”, por si entra dentro de tus inquietudes:


      Santiago Castillo
      • Hola Santiago, muchas gracias por el comentario.

        Yo no sé casi nada de matemáticas, pero le he perdido o le estoy perdiendo el miedo a pensar racionalmente también sobre ellas.

        Estos problemas son muy interesantes. Pienso que no debería ser difícil de entender, porque en realidad todo el cálculo se limita a sumar, restar, y hacer combinaciones. No hay cosas mágicas ahí.

        Pero para entenderlo primero tendríamos que comprender qué es una suma, qué es una multiplicación, qué es una potenciación, y qué es hallar una raíz.

        He hecho unos diagramas para mostrarlo visualmente, porque yo si no es visualmente no entiendo nada:

        Entonces, cuando sumamos 2 + 3, estamos formando un grupo de 5 juntando 2 y con 3.
        Cuando multiplicamos 2 x 3, estamos formando un grupo de 6 juntando 2 y 2 y 2 (o sea, repitiendo tres veces 2)
        Cuando elevamos 2 al cuadrado 2ˆ2, formamos un grupo de 4 juntando 2 y 2 ( o sea, repetimos dos veces 2)
        Y cuando elevamos 2 al cubo 2ˆ3, formamos primero un grupo de 2 y 2 (que sería el correspondiente a dos al cuadrado), y luego repetimos una vez más ese grupo. Así que lo que hacemos es juntar 4 y 4.
        Eso pasa cada vez que incrementamos la potencia. Por ejemplo 2ˆ4, supone que formamos un grupo de 2 y 2 (para 2ˆ2); luego repetimos ese grupo de 4 juntando 4 y 4 (para 2ˆ3); y luego repetimos ese grupo de 8 juntando 8 y 8 (para 2ˆ4). De manera que 2ˆ4 es 16.

        Lo primero que nos dice Google sobre los “cubos perfectos” es que “son los números que poseen raíces cúbicas exactas. 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375”

        Podían haberlos llamado “raíces enteras” o exactas, pero a alguien le dio un arrebato poético y le puso eso de los “cubos perfectos” que lo único que hace es despistarnos, porque no tiene que ver con la representación geométrica de la cifra.

        Pero por qué siguen ese orden y no otro los cubos perfectos? Por qué la raíz cúbica de 16 no nos da un resultado exacto sino que salen decimales, mientras que la raíz cúbica de 64 si es un número exacto?

        Bueno pues porque el resultado exacto depende de cómo se haya construido el número. Si es un número construido sobre la potencia de 2, la raíz cúbica nos va a dar decimales; y si es un número construído sobre la potencia de 3, la raíz cuadrada también nos va a dar decimales. Puede haber números que pueden construirse sobre la potencia de dos o la de tres, dando su raíz cuadrada y la cúbica un resultado exacto: por ejemplo 64. Lo que pasa aquí es que 64 podemos construirlo elevando 8 al cuadrado (8 x 8), y podemos construirlo elevando 4 al cubo, o sea 4 x 4 x 4.

        Si tomamos como ejemplo el caso de la raíz cuadrada perfecta de 256, vemos que podemos construir 256 con 16 x 16, o con 4 x 4 x 4 x 4, o con 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 (multiplicando 2 veces 16, 4 veces 4, o 8 veces 2).

        En cambio en otro caso como el de la raíz cúbica perfecta de 512, vemos que 512 sólo podemos construirlo con 8 x 8 x 8 (multiplicando 3 veces 8).

        Vemos que este comportamiento parece que se da en todos los casos. Las raíces cuadradas perfectas son de números que formamos multiplicándolos por sí mismos 2 veces (siendo 2 un número par), o multiplicándolos por sí mismos 4 veces u 8 veces (siendo 4 y 8 números pares formados con 2, lo que se llama múltiplos de 2). Mientras que las raíces cúbicas perfectas son de números que construimos multiplicándolos por sí mismos tres veces (siendo tres un número impar).

        Entonces si las secuencias de números del enlace de Gausianos tienen raíces cúbicas exactas, sabemos que es porque son números que se han construido sobre la potencia de tres. Y esa misma estructura es la que se tiene que mantener al ir agregando los otros números de la manera en que lo hace. Si te fijas las secuencias sólo repiten los números que son impares, 1, 7 y 11. El 0 y el 8 los deja como están inicialmente. Sin verlo claro, mi intuición me dice que a lo mejor por ahí viene el hecho de que se mantenga la estructura de la potenciación impar de 3. Y con esa pista es con la que seguiría investigando.

        Me gustaría investigarlo más y meterme a entender en profundidad teoría de números y cálculo (lo del cálculo diferencial – que no son más que infinitas aproximaciones – es algo a lo que también le tengo muchas ganas), pero los números enganchan y absorben mucho y yo no tengo demasiado tiempo.

        El problema d los cubos y cuadrados perfectos me parece relacionado también de alguna manera con el problema de los números primos, que son los que no están compuestos por otros anteriores (los matemáticos dicen que sólo son divisibles por ellos mismos y por 1). Por ejemplo el caso de 64 que se puede formar elevando a 2 el número 8 o elevando a 3 el número 4, no serían cuadrados ni cubos “primos”, si se les puede llamar así. Ya que tienen cuadrados y cubos perfectos.
        Mientras que 4, 16, o 256 si tendrían raíces cuadradas “primas”, y 8 y 512 tendrían raíces cúbicas “primas”.

        Un saludo muy cordial

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