Prime numbers in the Ulam square

About

In this page we shall collect the data and images related to our interest in the geometrical properties of the contiguos line segments formed by the primes in the Ulam square.

Leszek J Chmielewski and Arkadiusz Orłowski
Authors

Copyright

Copyright (C) Leszek J Chmielewski and Arkadiusz Orłowski.

Only non-commercial use of these dat is allowed.

The precondition for using these data is that you cite this web page "www.lchmiel.pl/primes" and the publications selected by you from this literature list.

Selected data

Each of the sets in the table below for the Ulam square of the specified size contains:

• Text files:
  1. file with information on the size of the square, numbers of primes, maximum prime;
  2. files with listings of segments of length from 4 to 16 pixels (empty files if there are no segments with a length).
• Images:
  1. Ulam spiral;
  2. images with visualizations of segments of length from 4 to 16 pixels (those with nonzero number of segments); for some larger squares the images can be unavailable.

Table of data sets

Size of Ulam square	Size of direction table	Number of points	Number of primes	Largest prime	Data file	Remarks
301×301	[0,10]×[-10,10]	...	...	...	...
501×501	[0,10]×[-10,10]	251001	22115	250993	dowlnoad
1001×1001	[0,10]×[-10,10]	...	...	...	...
3001×3001	[0,10]×[-10,10]	...	...	...	...
5001×5001	[0,10]×[-10,10]	25010001	1566540	25009991	download	Images of segments are currently not available.
9001×9001	[0,5]×[-5,5]	81018001	4725457	81017969	download	Images of segments are currently not available.
10001×10001	[0,3]×[-3,3]	100020001	5762536	100019977	download	Images of segments are currently not available.

Literature

1. L. J. Chmielewski, M. Janowicz, G. Gawdzik and A. Orłowski. Contiguous line segments in the Ulam spiral: Experiments with larger numbers. In Artificial Intelligence and Soft Computing: Proc. Int. Conf. ICAISC 2017, pages 463-472, Zakopane, Poland, 11-15 Apr 2017. LNAI series Vol. 10245, Springer. (doi:10.1007/978-3-319-59063-9_41) (PDF)
2. L. J. Chmielewski, A. Orłowski, and G. Gawdzik. Segment counting versus prime counting in the Ulam square. In Proc. 9th Asian Conference on Intelligent Information and Database Systems ACIIDS 2017, Part II, pages 227-236, Kanazawa, Japan, 3-5 Apr 2017. LNAI series Vol. 10192, Springer. (doi:10.1007/978-3-319-54430-4_22) (PDF)
3. L.J. Chmielewski, A. Orłowski and M. Janowicz. A study on directionality in the Ulam square with the use of the Hough Transform. In Hard and Soft Computing for Artificial Intelligence, Multimedia and Security: Proc. Conf. ACS 2016, pages 81-90, Międzyzdroje, Poland, Oct 19-21, 2016. Advances in Intelligent Systems and Computing series Vol. 534, Springer, 2016. (doi:10.1007/978-3-319-48429-7_8) (PDF)
4. L.J. Chmielewski, A. Orłowski. Finding line segments in the Ulam square with the Hough transform. In Computer Vision and Graphics: Proc. Int. Conf. Computer Vision and Graphics ICCVG 2016, pages 617-626, Warsaw, Poland, Sept 19-21, 2016. LNCS series Vol. 9972, Springer. (doi:10.1007/978-3-319-46418-3_55) (PDF)
5. L.J. Chmielewski, A. Orłowski. Hough transform for lines with slope defined by a pair of co-primes. Machine Graphics & Vision, 22(1-4):17-25, 2013 [open access].