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

Leo Depuydt^{*}

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References

[1] L. E. Dickson, “History of the Theory of Numbers, I: Divisibility and Primality,” G. E. Stechert & Co., New York, 1934.

[2] J. Derbyshire, “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics,” Joseph Henry Press, Washington DC, 2003.

[3] M. du Sautoy, “The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics,” Harper Collins Publishers, New York, 2003.

[4] D. Wells, “Prime Numbers: The Most Mysterious Figures in Math,” John Wiley & Sons, Inc., Hoboken, NJ, 2005.

[5] L. Depuydt, “The Mathematical and Physical Theory of Rational Human Intelligence: Complete Empirical-Digital Properties; Full Electrochemical-Mechanical Model (Part I: Mathematical Foundations),” Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 3, No. 5 (August 2013), pp. 491-561; bibliographical references to earlier publications pertaining to the larger project are listed there. 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. Vagliente, Boston Studies in the Philosophy of Science, Vol. 191, Kluwer Academic Publishers, Dordrecht/Boston/London, p. xii.

[27] J.[-]L. Lagrange, Analytical Mechanics, Translated and edited by A. Boissonnade and V. N. Vagliente, Boston Studies in the Philosophy of Science, Vol. 191, Kluwer Academic Publishers,

Dordrecht/Boston/London, 1997, p. 28.

[28] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960, pp. 273-296.

[29] At www.jamestanton.com.

[30] W. A. Whitworth, Choice and Chance, with One Thousand Exercises, Hafner Publishing, New York and London, 1965, reprint of the fifth edition of 1901, pp. 93-96.

[31] J. E. Pommersheim, T. K. Marks, and E. L. Flapan, Number Theory: A Lively Introduction with Proofs, Applications, and Stories, John Wiley & Sons, Inc., Hoboken, NJ, 2010, p. 652.

[32] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960, p. 276.

[33] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960, p. 274.

[34] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960, p. 277.

[35] Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, p. 219.

[36] E. Beltrami, “Saggio di interpretazione della geometria non-euclidea,” Giornale di matematiche, Vol. 6, 1868, pp. 284-312 (a French translation appeared in Annales scientifiques de l’école Normale Supérieure, Series 1, Vol. 6, 1869, pp. 251-288).

[37] J. Hoüel, “Note sur l’impossibilité de démontrer par une construction plane le principe de la théorie des parallèles dit Postulatum d’Euclide,” Giornale di matematiche, Vol. 8, 1870, pp. 84-89.

[38] E. Schroder, “Vorlesungen über die Algebra der Logik (Exakte Logik),” Vol. 1, J. C. Hinrichs, Leipzig, 1890, p. 288.

[39] I. (Iohannes) L. Heiberg, Ed., “Euclidis Elementa,” Vol. II, B. G. Teubner, Leipzig, 1884, p. 186.

[40] E. S. Stamatis, Ed., after I. L. Heiberg, Ed., “Euclidis Elementa,” Vol. II, B. G. Teubner, Leipzig, 1970, p. 104.

[41] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 2, Dover Publications, New York, 1956, p. 277.

[42] Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 2, Dover Publications, New York, 1956, p. 279.

[43] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, p. 155.

[44] Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, pp. 202-220.

[45] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, p. 155.

[46] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, p. 154.

[47] Translation by Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 1, Dover Publications, New York, 1956, p. 312 and p. 342.

[48] J.-L. Lagrange, “Démonstration d’un théorème nouveau concernant les nombres premiers,” In: J.-A. Serret, Ed., Oeuvres de Lagrange, Vol. 3, Gauthier-Villars, Paris, 1869, pp. 425-438, at p. 425.

[49] J.-L. Lagrange, “Démonstration d’un théorème nouveau concernant les nombres premiers,” In: J.-A. Serret, Ed., Oeuvres de Lagrange, Vol. 3, Gauthier-Villars, Paris, 1869, pp. 425-438.

[50] L. E. Dickson, “History of the Theory of Numbers, I: Divisibility and Primality,” G. E. Stechert & Co., New York, 1934, p. 426.

[51] Th. L. Heath, “Euclid: The Thirteen Books of the Elements,” Vol. 2, Dover Publications, New York, 1956, p. 413.

[52] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, Oxford, 1960, p. 12.

[53] J. E. Pommersheim, T. K. Marks, and E. L. Flapan, Number Theory: A Lively Introduction with Proofs, Applications, and Stories, John Wiley & Sons, Inc., Hoboken, N. J., 2010, p. 105.

[54] L. E. Dickson, “History of the Theory of Numbers, I: Divisibility and Primality,” G. E. Stechert & Co., New York, 1934, p. 413.

[55] D. Wells, “Prime Numbers: The Most Mysterious Figures in Math,” John Wiley & Sons, Inc., Hoboken, NJ, 2005, p. 166.

[56] A. W. F. Edwards, “Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea,” Johns Hopkins University Press, Baltimore, 2002.

[57] L. Euler, “Elements of Algebra,” Springer Verlag, New York, Berlin, Heidelberg, and Tokyo, 1984 (reprint of the English translation of 1840), p. 114.

[58] L. Depuydt, “The Mathematical and Physical Theory of Rational Human Intelligence,” Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 3, No. 5 (August 2013), pp. 491-561, at p. 577.

[59] Ch. Burney, “A General History of Music,” Vol. IV, Printed for the Author, London, 1789, p. 557.

[60] A. E. M. Grétry, “Mémoires ou essais sur la musique,” Vol. I, Printer of the Republic, Paris, Year 5 [of the French Republic], p. 424.

[61] G. Radiciotti, “G. B. Pergolesi: Vita, opere ed influenza su l’arte (con multi esempi musicali ed illustrazioni),” Edizione “Musica,” Rome, 1910.

[62] L. Depuydt, “The Mathematical and Physical Theory of Rational Human Intelligence,” Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 3, No. 5 (August 2013), pp. 491-561, passim.

[63] B. Mahon, “The Man Who Changed Everything,” John Wiley & Sons, Ltd, Chichester, West Sussex, 2004, especially at p. xi and pp. 176-185.

[64] B. Mahon, “The Man Who Changed Everything,” John Wiley & Sons, Ltd, Chichester, West Sussex, 2004, p. 176.

[65] C. B. Boyer, “A History of Mathematics,” Second Edition, John Wiley & Sons, Inc., New York, 1989, p. 490.

[66] J.[-]L. Lagrange, Analytical Mechanics, Translated and edited by A. Boissonnade and V. N. Vagliente, Boston Studies in the Philosophy of Science, Vol. 191, Kluwer Academic Publishers, Dordrecht/Boston/London, 1997, p. 28.

[67] Th. Hailperin, “Boole’s Logic and Probability” (First Edition), Studies in Logic and the Foundations of Mathematics, Vol. 85, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1976.

[68] Th. Hailperin, “Boole’s Logic and Probability,” Second edition, Revised and enlarged, Studies in Logic and the Foundations of Mathematics, Vol. 85, North-Holland Publishing Company, Amsterdam, New York, Oxford, and Tokyo, 1986.

[69] G. C. Smith, “The Boole-De Morgan Correspondence,” Clarendon Press, Oxford, 1982.

[70] D. MacHale, “George Boole: His Life and Work,” Boole Press, Dublin, 1985.

[71] I. Grattan-Guinness, “The Correspondence between George Boole and Stanley Jevons, 1863-1864,” History and Philosophy of Logic, Vol. 12, 1991, pp. 15-35. http://dx.doi.org/10.1080/01445349108837175

[72] I. Grattan-Guinness and G. Bornet, Eds., “George Boole: Selected Manuscripts on Logic and Its Philosophy,” Science Networks Historical Studies, Vol. 20, Birkhauser Verlag, Basel, Boston, and Berlin, 1997.?

[73] J. Gasser, Ed., “A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,” Kluwer Academic Publishers, Dordrecht, Boston, and London, 2000.

[74] M. H. Paymer and H. W. Williams, “Giovanni Battista Pergolesi: A Guide to Research,” Garland Composer Resources Manuals, Vol. 26, Garland Publishing Inc., New York and London, 1989.

[75] A. E. M. Grétry, “Mémoires ou essais sur la musique,” Vol. I, Printer of the Republic, Paris, Year 5 [of the French Republic], p. 424.

[76] Ch. Burney, “A General History of Music,” Vol. IV, Printed for the Author, London, 1789, p. 556.

[77] G. Radiciotti, “G. B. Pergolesi: Vita, opere ed influenza su l’arte (con multi esempi musicali ed illustrazioni),” Edizione “Musica,” Rome, 1910, p. 275.

[78] M. E. Paymer and H. W. Williams, “Giovanni Battista Pergolesi: A Guide to Research,” Garland Composer Resources Manuals, Vol. 26, Garland Publishing Inc., New York and London, 1989, p. 84.

[79] M. E. Paymer and H. W. Williams, “Giovanni Battista Pergolesi: A Guide to Research,” Garland Composer Resources Manuals, Vol. 26, Garland Publishing Inc., New York and London, 1989, p. 98.

[80] M. E. Paymer and H. W. Williams, “Giovanni Battista Pergolesi: A Guide to Research,” Garland Composer Resources Manuals, Vol. 26, Garland Publishing Inc., New York and London, 1989, p. 105.

[81] M. E. Paymer and H. W. Williams, “Giovanni Battista Pergolesi: A Guide to Research,” Garland Composer Resources Manuals, Vol. 26, Garland Publishing Inc., New York and London, 1989, p. 96.

[82] A. E. M. Grétry, “Mémoires ou essais sur la musique,” Vol. I, Printer of the Republic, Paris, Year 5 [of the French Republic], pp. 426-427.

[83] Ch. Burney, “A General History of Music,” Vol. IV, Printed for the Author, London, 1789, pp. 551-552.

[84] G. Radiciotti, “G. B. Pergolesi: Vita, opere ed influenza su l’arte (con multi esempi musicali ed illustrazioni),” Edizione “Musica,” Rome, 1910, p. 27.

[85] G. Radiciotti and A.-E. Cherbulliez, Ed., “Giovanni Battista Pergolesi: Leben und Werk,” Pan-Verlag, Zürich, 1954, p. 49.

[86] G. Radiciotti and A.-E. Cherbulliez, Ed., “Giovanni Battista Pergolesi: Leben und Werk,” Pan-Verlag, Zürich, 1954, see index at “Leo, L.”.