The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable—With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence
Author(s)
Leo Depuydt*
ABSTRACT
The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes—that is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and so on. The main conclusion revolves entirely around two points. First, on the one hand, it is shown that the prime sequence exhibits an extremely high level of organization. But second, on the other hand, it is also shown that the clearly detectable organization of the primes is ultimately beyond human comprehension. This conclusion runs radically counter and opposite—in regard to both points—to what may well be the default view held widely, if not universally, in current theoretical mathematics about the prime sequence, namely the following. First, on the one hand, the prime sequence is deemed by all appearance to be entirely random, not organized at all. Second, on the other hand, all hope has not been abandoned that the sequence may perhaps at some point be grasped by human cognition, even if no progress at all has been made in this regard. Current mathematical research seems to be entirely predicated on keeping this hope alive. In the present paper, it is proposed that there is no reason to hope, as it were. According to this point of view, theoretical mathematics needs to take a drastic 180-degree turn. The manner of demonstration that will be used is direct and empirical. Two key observations are adduced showing, 1), how the prime sequence is highly organized and, 2), how this organization transcends human intelligence because it plays out in the dimension of infinity and in relation to π. The present paper is part of a larger project whose design it is to present a complete and final mathematical and physical theory of rational human intelligence. Nothing seems more self-evident than that rational human intelligence is subject to absolute limitations. The brain is a material and physically finite tool. Everyone will therefore readily agree that, as far as reasoning is concerned, there are things that the brain can do and things that it cannot do. The search is therefore for the line that separates the two, or the limits beyond which rational human intelligence cannot go. It is proposed that the structure of the prime sequence lies beyond those limits. The contemplation of the prime sequence teaches us something deeply fundamental about the human condition. It is part of the quest to Know Thyself.
KEYWORDS
Absolute Limitations of Rational Human Intelligence, Analytic Number Theory, Aristotle’s Fundamental Axiom of Thought, Euclid’s Fifth Postulate, Euclid on Numbers, Euclid on Primes, Euclid’s Proof of the Primes’ Infinitude, Euler’s Infinite Prime Product, Euler’s Infinite Prime Product Equation, Euler’s Product Formula, Gödel’s Incompleteness Theorem, Goldbach’s Conjecture, Lagrange’s Proof of Wilson’s Theorem, Number Theory, Partition, Partition Numbers, Prime Numbers (Primes), Prime Sequence (Sequence of the Prime Numbers), Rational Human Intelligence, Rational Thought and Language, Riemann’s Hypothesis, Riemann’s Zeta Function, Wilson’s Theorem
Absolute Limitations of Rational Human Intelligence, Analytic Number Theory, Aristotle’s Fundamental Axiom of Thought, Euclid’s Fifth Postulate, Euclid on Numbers, Euclid on Primes, Euclid’s Proof of the Primes’ Infinitude, Euler’s Infinite Prime Product, Euler’s Infinite Prime Product Equation, Euler’s Product Formula, Gödel’s Incompleteness Theorem, Goldbach’s Conjecture, Lagrange’s Proof of Wilson’s Theorem, Number Theory, Partition, Partition Numbers, Prime Numbers (Primes), Prime Sequence (Sequence of the Prime Numbers), Rational Human Intelligence, Rational Thought and Language, Riemann’s Hypothesis, Riemann’s Zeta Function, Wilson’s Theorem
Cite this paper
Depuydt, L. (2014) The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable—With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence. Advances in Pure Mathematics, 4, 400-466. doi: 10.4236/apm.2014.48051.
Depuydt, L. (2014) The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable—With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence. Advances in Pure Mathematics, 4, 400-466. doi: 10.4236/apm.2014.48051.
References
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The following typos appear in this article (references are to page, column, and line): (506,a,11) for “or modern” read “of modern”; (513,a,30) for “chose” read “choose”; (515,a,10 counting from bottom) for “and” read “an”; (546,b,19) for “incrase” read “increase”; (552,b,30) for “describe” read “described”; (554,a,34) for “Maxwell’s” read “Maxwell” (555,b,37) for “interested” read “interest”; (559,a,note 14,1]) for “P. E. B. Jourdain” read “Ph. E. B. Jourdain”; (559,b,note 67,3). Also, at 553,a,3, for “below” read “in later articles.” [6] As for CerDi, see L. Depuydt, “The Other Mathematics,” Gorgias Press, Piscataway, NJ, 2008, especially at pp. 79-95 and pp. 307-321. [7] As for SupDi, see L. 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The following typos appear in this article (references are to page, column, and line): (506,a,11) for “or modern” read “of modern”; (513,a,30) for “chose” read “choose”; (515,a,10 counting from bottom) for “and” read “an”; (546,b,19) for “incrase” read “increase”; (552,b,30) for “describe” read “described”; (554,a,34) for “Maxwell’s” read “Maxwell” (555,b,37) for “interested” read “interest”; (559,a,note 14,1]) for “P. E. B. Jourdain” read “Ph. E. B. Jourdain”; (559,b,note 67,3). Also, at 553,a,3, for “below” read “in later articles.” [6] As for CerDi, see L. Depuydt, “The Other Mathematics,” Gorgias Press, Piscataway, NJ, 2008, especially at pp. 79-95 and pp. 307-321. [7] As for SupDi, see L. Depuydt, “Zur unausweichlichen Digitalisierung der Sprachbetrachtung: ‘Allein’, ‘anderer’, ‘auch’, ‘einziger’, ‘(seiner)seits’, und ‘selbst’ als digitales Wortfeld im Agyptisch-Koptischen und im Allgemeinen [On the Unavoidable Digitalization of Language Analysis: ‘Alone’, ‘Other’, ‘Also’, ‘Only’, ‘On (His) Part’, and ‘Self’ as Digital Semantic Field in Egyptian-Coptic and in General],” In: A. I. Blobaum, K. Butt, and I. Kohler, Eds., Lexical Fields, Semantics and Lexicography, Aegyptiaca Monasteriensia, Vol. 7, Shaker Verlag, Aachen, 2011, pp. 5-38. [8] www.oeis.org. Search for “A231273” and “A231327.” [9] M. du Sautoy, “The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics,” Harper Collins Publishers, New York, 2003, p. 314. [10] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 2, Dover Publications, New York, 1956, p. 278. [11] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, p. 153. [12] J. Derbyshire, “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics,” Joseph Henry Press, Washington DC, 2003, p. 9. [13] J. Derbyshire, “Prime Obsession,” Joseph Henry Press, Washington DC, 2003, pp. 63-65. [14] J. Derbyshire, “Prime Obsession,” Joseph Henry Press, Washington DC, 2003, pp. 102-107. [15] B. Riemann, “über die Anzahl der Primzahlen unter einer gegebenen Grosse,” Monatsberichte der Koniglichen Preussischen Akademie der Wissenschaften zu Berlin: Aus dem Jahre 1859, F. Dümmler, Berlin, 1860, pp. 671-680.? [16] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960, p. 4. [17] http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/03/02 (accessed last on October 31, 2013). [18] L. Euler, “Elements of Algebra,” Springer Verlag, New York, Berlin, Heidelberg, and Tokyo, 1984 (reprint of the English translation of 1840 by J. Hewlett of, [1], the French translation by J. Bernoulli (III) of the original German edition of 1770 and of, [2], the French additions by J.-L. Lagrange, with a preface by C. Truesdell that earlier appeared in H. E. Pagliaro, Ed., “Irrationalism in the Eighteenth Century,” The Press of Case Western Reserve University, Cleveland and London, 1972, pp. 51-95), p. 92. [19] D. Wells, “Prime Numbers: The Most Mysterious Figures in Math,” John Wiley & Sons, Inc., Hoboken, NJ, 2005, p. 69. [20] B. Riemann, “über die Anzahl der Primzahlen unter einer gegebenen Grosse,” Monatsberichte der Koniglichen Preussischen Akademie der Wissenschaften zu Berlin: Aus dem Jahre 1859, F. Dümmler, Berlin, 1860, pp. 671-680, at p. 671. [21] J. Derbyshire, “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics,” Joseph Henry Press, Washington DC, 2003, p. 327. [22] P. J. Nahin, “An Imaginary Tale: The Story of ,” Princeton University Press, Princeton and London, p. 173. [23] http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/03/02 (accessed last on October 31, 2013). [24] For the Greek text, see Aristotle, “Metaphysics Books I-IX,” with an English translation by H. Tredennick, Loeb Classical Library, Vol. 271, Harvard University, Cambridge, MA, 1933, p. 160 and p. 162. [25] M. Dehn, “Beziehungen zwischen der Philosophie und der Grundlegung der Mathematik im Altertum,” Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Vol. B.4.1, Julius Springer, Berlin, 1937, pp. 1-28, at p. 8. [26] J.[-]L. Lagrange, Analytical Mechanics, Translated from the Mécanique analytique, no[u]velle edition of 1811, Translated and edited by A. Boissonnade and V. N. 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