Prime constellations in triangles with binomial coefficient congruences

Larry Ericksen

Acta Mathematica Universitatis Ostraviensis (2009)

  • Volume: 17, Issue: 1, page 67-80
  • ISSN: 1804-1388

Abstract

top
The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by the diagonal representation of Ericksen. Primes of linear and polynomial forms are identified from congruences of their associated binomial coefficients. This method of primality testing is extended to triangle elements created from q -binomial or Gaussian coefficients, using congruences with cyclotomic polynomials as a modulus. We apply Kummer’s method of p -ary representation to binomial coefficient congruences to find prime constellations. Aside from their capacity to find prime numbers in binomial coefficient triangles, congruences are used to identify prime properties of composite numbers, represented as distinct prime factors or as prime pairs.

How to cite

top

Ericksen, Larry. "Prime constellations in triangles with binomial coefficient congruences." Acta Mathematica Universitatis Ostraviensis 17.1 (2009): 67-80. <http://eudml.org/doc/35198>.

@article{Ericksen2009,
abstract = {The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by the diagonal representation of Ericksen. Primes of linear and polynomial forms are identified from congruences of their associated binomial coefficients. This method of primality testing is extended to triangle elements created from $q$-binomial or Gaussian coefficients, using congruences with cyclotomic polynomials as a modulus. We apply Kummer’s method of $p$-ary representation to binomial coefficient congruences to find prime constellations. Aside from their capacity to find prime numbers in binomial coefficient triangles, congruences are used to identify prime properties of composite numbers, represented as distinct prime factors or as prime pairs.},
author = {Ericksen, Larry},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {Binomial coefficient; congruence; cyclotomic polynomial; Kummer’s theorem; Gaussian binomial coefficient; Pascal’s triangle; prime constellation; primality test; Binomial coefficient; congruence; cyclotomic polynomial; Kummer's theorem; Gaussian binomial coefficient; Pascal's triangle; prime constellation; primality test},
language = {eng},
number = {1},
pages = {67-80},
publisher = {University of Ostrava},
title = {Prime constellations in triangles with binomial coefficient congruences},
url = {http://eudml.org/doc/35198},
volume = {17},
year = {2009},
}

TY - JOUR
AU - Ericksen, Larry
TI - Prime constellations in triangles with binomial coefficient congruences
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2009
PB - University of Ostrava
VL - 17
IS - 1
SP - 67
EP - 80
AB - The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by the diagonal representation of Ericksen. Primes of linear and polynomial forms are identified from congruences of their associated binomial coefficients. This method of primality testing is extended to triangle elements created from $q$-binomial or Gaussian coefficients, using congruences with cyclotomic polynomials as a modulus. We apply Kummer’s method of $p$-ary representation to binomial coefficient congruences to find prime constellations. Aside from their capacity to find prime numbers in binomial coefficient triangles, congruences are used to identify prime properties of composite numbers, represented as distinct prime factors or as prime pairs.
LA - eng
KW - Binomial coefficient; congruence; cyclotomic polynomial; Kummer’s theorem; Gaussian binomial coefficient; Pascal’s triangle; prime constellation; primality test; Binomial coefficient; congruence; cyclotomic polynomial; Kummer's theorem; Gaussian binomial coefficient; Pascal's triangle; prime constellation; primality test
UR - http://eudml.org/doc/35198
ER -

References

top
  1. Bondarenko, B. A., Generalized Pascal triangles and pyramids: Their fractals, graphs, and applications, pp. 22–23, 58. Santa Clara, CA: The Fibonacci Association, 1993 (1993) Zbl0792.05001
  2. Caldwell, C., The Prime Pages, http://primes.utm.edu/notes/proofs/Wilsons.html 
  3. Crandall, R., Pomerance, C., Prime Numbers. A Computational Perspective., Springer-Verlag, New York, 2001 (2001) MR1821158
  4. Dickson, L. E., A New Extension of Dirichlet’s Theorem on Prime Numbers, Messenger Math. 33, 1904, p. 155–161 (1904) 
  5. Dilcher, K., Stolarsky, K. B., 10.2307/30037570, Amer. Math. Monthly 112, 2005, p. 673–681 (2005) Zbl1159.11303MR2167768DOI10.2307/30037570
  6. Dirichlet, P. G. L., Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abhand. Ak. Wiss. Berlin 48, 1837 (1837) 
  7. Ericksen, L., 10.2478/s12175-009-0122-7, Math. Slovaca 59(3), 2009, p. 261–274 (2009) Zbl1209.11012MR2505805DOI10.2478/s12175-009-0122-7
  8. Ericksen, L., Iterated Digit Sums, Recursions and Primality, Acta Mathematica Universitatis Ostraviensis 14, 2006, p. 27–35 (2006) Zbl1148.11007MR2298910
  9. Ericksen, L., Primality testing and prime constellations, Šiauliai Math. Semin. 3(11), 2008, p. 61–77 (2008) Zbl1163.11007MR2543450
  10. Harborth, H., Über die teilbarkeit im Pascal-Dreieck, Math.-Phys Semesterber 22, 1975, p. 13–21 (1975) Zbl0296.10009MR0384676
  11. Harborth, H., 10.1007/BF01224673, Arch. Math. 27(3), 1976, p. 290–294 (1976) Zbl0325.10004MR0417038DOI10.1007/BF01224673
  12. Harborth, H., Prime number criteria in Pascal’s triangle, J. London Math. Soc. (2) 16, 1977 p. 184–190 (1977) Zbl0371.10002MR0476623
  13. Hudson, R. H., Williams, K. S., A divisibility property of binomial coefficients viewed as an elementary sieve, Internat. J. Math. & Math. Sci. 4(4), 1981, p. 731–743 (1981) Zbl0479.10005MR0663657
  14. Kummer, E. E., Über die Erganzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44, 1852, p. 93–146 (1852) 
  15. Mann, H. B., Shanks, D., A necessary and sufficient condition for primality and its source, J. Combinatorial Theory (A) 13, 1972, p. 131–134 (1972) Zbl0239.10010MR0306098
  16. Pascal, B., Traité du triangle arithmétique, avec quelques autres petits traités sur la mêmes matières, G. Desprez, Paris, 1654, Oeuvres completes de Blaise Pascal, T. 3, 1909, p. 433–593 (1909) 
  17. Robin, G., Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63, 1984, p. 187–213 (1984) Zbl0516.10036MR0774171
  18. Schinzel, A., Sierpinski, W., Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4, 1958, p. 185–208, Erratum 5, 1959, p. 259 (1958) Zbl0082.25802MR0106202
  19. Weisstein, E.W., Distinct Prime Factors, http://mathworld.wolfram.com/DistinctPrimeFactors.html 
  20. Weisstein, E.W., Prime Constellation, http://mathworld.wolfram.com/PrimeConstellation.html 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.