Proof the Skewes’ number is not an integer using lattice points and tangent line
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References
- [1] Garey M. R., Johnson D. S. (1990). Computers and Intractability – A Guide to the Theory of NP-Completeness. NY, USA: W. H. Freeman & Co., ISBN 0716710455.
- [2] Riesel H. (2012). Prime Numbers and Computer Methods for Factorization. Springer: New York, 2nd ed., 482 p., ISBN 978-0-8176-8298-9, DOI 10.1007/978-0-8176-8298-9.
- [3] Coutinho S. C. (1999). The Mathematics of Ciphers: Number Theory and RSA Cryptography. 1st ed., 198 p., Brazil: A. K. Peters, ISBN: 9781568810829.
- [4] Kleinjung T. et al. (2010). Factorization of a 768-Bit RSA Modulus. In: Advances in Cryptology – CRYPTO 2010. Lecture Notes in Computer Science, vol 6223. Springer, Berlin, Heidelberg, DOI: 10.1007/978-3-642-14623-7_18.
- [5] Sittinger, B. D. (2010). The probability that random algebraic integers are relatively r-prime. In. Journal of Number Theory, Vol. 130, No. 1, pp. 164-171., DOI 10.1016/j.jnt.2009.06.008.
- [6] Ďuriš V. (2020). Solving Some Special Task for Arithmetic Functions and Perfect Numbers. In. 19th Conference on Applied Mathematics: proceeding, Bratislava: STU, 4th -6th of February, 2020, pp. 374-383, ISBN 978-80-227-4983-1.
- [7] Lehman R. (1974). Factoring large integers. In: Mathematics of Computation, Vol. 28, No. 126, p. 637-646.
- [8] Koblitz N. (1994). A Course in Number Theory and Cryptography. 2nd ed., New York: Springer-Verlag, ISBN 0387942939.
- [9] Menezes J. A., Oorschot P. C., Vanstone S. A. (1997). Applied Cryptography. New York: CRC, ISBN 0-8493-8523-7.
- [10] Ribenboim P. (2004). The Little Book of Big Primes. USA, NY: Springer-Verlag, 368 p., ISBN 978-0-387-21820-5.
- [11] Smith D. E. (1958). History of Mathematics, Vol. I. 618p., 1st ed., US: Dover Publications.
- [12] Smith D. E. (1958). History of Mathematics, Vol. II. 736p., 1st ed., US: Dover Publications.
- [13] Legendre A. M. (1798). Essai sur la théorie de Nombres, 1st ed., Paris.
- [14] Legendre A. M. (1808). Essai sur la théorie de Nombres, 2nd ed., Paris.
- [15] Pintz J. (1980). On Legendre’s prime number formula. In. The American Mathematical Monthly, Vol. 87., No. 9., pp. 733-735.
- [16] Mazur B. (2016). Prime Numbers and the Riemann Hypothesis. 1st ed., UK: Cambridge University Press, 150 p., ISBN: 978-1107499430.
- [17] Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques.In. Bulletin de la Société Mathématique de France, No. 24, pp. 199-220.
- [18] De la Vallée Poussin C. J. (1896). Recherches analytiques de la théorie des nombres premiers. In. Annales de la Societe Scientifique de Bruxelles No. 20, pp. 183-256.
- [19] Williams H. P. (2007). Stanley Skewes and the Skewes number. In. Journal of the Royal Institution of Cornwall, ISSN 0968-5396, pp. 70-75.
- [20] Abramowitz M., Stegun I. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Revised ed., USA, Dover Publications, 1046 p., ISBN 978-0486612720.
- [21] Kanwal R. P. (1996). Linear Integral Equations: Theory and technique. 2nd ed., USA Boston: Birkhäuser, 318 p., ISBN 978-0817639402:.
- [22] Littlewood J. E. (1914). Sur la distribution des nombres premiers. In. Comptes Rendus de l’Acad. Sci. Paris, Vol. 158, pp. 1869-1875.
- [23] Skewes S. (1933). On the difference π(x) − li(x). In. Journal of the London Mathematical Society, Vol. s1-8, No. 4, pp. 277–283, DOI: 10.1112/jlms/s1-8.4.277.
- [24] Skewes S. (1955). On the difference π(x) − li(x). In. Proceedings of the London Mathematical Society, Vol. s3-5, No. 1, pp. 48–70, DOI: 10.1112/plms/s3-5.1.48.
- [25] Ďuriš V., Šumný T. (2019). Diophantine Geometry in Space E2 and E3. In. TEM Journal, ISSN 2217-8309, Vol. 8, No. 1, pp. 78-81, DOI 10.18421/TEM81-10.
- [26] Rektorys K. (1968). Overview of applied mathematics. Prague: SNTL, Language: Czech.
- [27] Bronštejn I. N., Semendjaev K. A. (1961). Handbook of mathematics for engineers and for students at technical universities. Bratislava: SVTL, Language: Slovak, transl. from 8th. Rus. ed.
Language: English
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Submitted on: Sep 1, 2021
Published on: Dec 30, 2021
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