Contiguous Line Segments in the Ulam Spiral: Experiments with Larger Numbers

  • Leszek J. ChmielewskiEmail author
  • Maciej Janowicz
  • Grzegorz Gawdzik
  • Arkadiusz Orłowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10245)


In our previous papers we have investigated the directional structure and the numbers of straight line segments in the Ulam spiral. Our tests were limited to primes up to \(25\,009\,991\) due to memory limits. Now we have results for primes up to about \(10^9\) for the previously used directional resolution, and for the previous maximum number but with greatly increased directional resolution. For the extended resolution, new long segments have been found, among them the first one with 14 points. For larger numbers and the previous resolution, the new segments having up to 13 points were found, but the longest one is still the one with 16 points. It was confirmed that the relation of the number of segments of various lengths to the corresponding number of primes for a given integer, for large numbers, is close to linear in the double logarithmic scale.


Ulam spiral Ulam square Prime numbers Line segments Number of segments Large numbers 


  1. 1.
    Caldwell, C.K. (ed.): The Prime Pages (2016). Accessed 15 Oct 2016
  2. 2.
    Chmielewski, L.J., Orłowski, A.: Hough transform for lines with slope defined by a pair of co-primes. Mach. Graph. Vis. 22(1/4), 17–25 (2013). Google Scholar
  3. 3.
    Chmielewski, L.J., Orłowski, A.: Finding line segments in the Ulam square with the Hough transform. In: Chmielewski, L.J., Datta, A., Kozera, R., Wojciechowski, K. (eds.) ICCVG 2016. LNCS, vol. 9972, pp. 617–626. Springer, Cham (2016). doi: 10.1007/978-3-319-46418-3_55 CrossRefGoogle Scholar
  4. 4.
    Chmielewski, L.J., Orłowski, A.: Prime numbers in the Ulam square (2016). Accessed 14 Oct 2016
  5. 5.
    Chmielewski, L.J., Orłowski, A., Gawdzik, G.: Segment counting versus prime counting in the Ulam square. In: Nguyen, N.T., Tojo, S., Nguyen, L.M., Trawiński, B. (eds.) ACIIDS 2017. LNCS, vol. 10192, pp. 227–236. Springer, Cham (2017). doi: 10.1007/978-3-319-54430-4_22 CrossRefGoogle Scholar
  6. 6.
    Chmielewski, L.J., Orłowski, A., Janowicz, M.: A study on directionality in the Ulam square with the use of the Hough transform. In: Kobayashi, S., Piegat, A., Pejaś, J., El Fray, I., Kacprzyk, J. (eds.) ACS 2016. AISC, vol. 534, pp. 81–90. Springer, Cham (2017). doi: 10.1007/978-3-319-48429-7_8 CrossRefGoogle Scholar
  7. 7.
    GeMir: Quadratic polynomials describe the diagonal lines in the Ulam-Spiral. Mathematics Stack Exchange, 15 July 2015. Accessed 28 Jan 2017
  8. 8.
    PrimeCrank: Ulam spiral explained (sort of...). Prime Numbers on, 22 February 2012. Accessed 28 Jan 2017
  9. 9.
    Stein, M.L., Ulam, S.M., Wells, M.B.: A visual display of some properties of the distribution of primes. Am. Math. Mon. 71(5), 516–520 (1964). doi: 10.2307/2312588 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Weisstein, E.W.: Prime counting function. From MathWorld-A Wolfram Web Resource (2016). Accessed 15 Oct 2016
  11. 11.
    Weisstein, E.W.: Prime spiral. From MathWorld-A Wolfram Web Resource (2016). Accessed 15 Oct 2016

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Leszek J. Chmielewski
    • 1
    Email author
  • Maciej Janowicz
    • 1
  • Grzegorz Gawdzik
    • 1
  • Arkadiusz Orłowski
    • 1
  1. 1.Faculty of Applied Informatics and Mathematics – WZIMWarsaw University of Life Sciences – SGGWWarsawPoland

Personalised recommendations