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Binomial coefficient; congruence; cyclotomic polynomial; Kummer’s theorem; Gaussian binomial coefficient; Pascal’s triangle; prime constellation; primality test
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.
[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 Zbl 0792.05001
[2] Caldwell, C.: The Prime Pages.
[3] Crandall, R., Pomerance, C.: Prime Numbers. A Computational Perspective. Springer-Verlag, New York, 2001 MR 1821158
[4] Dickson, L. E.: A New Extension of Dirichlet’s Theorem on Prime Numbers. Messenger Math. 33, 1904, p. 155–161
[5] Dilcher, K., Stolarsky, K. B.: A Pascal-Type Triangle Characterizing Twin Primes. Amer. Math. Monthly 112, 2005, p. 673–681 DOI 10.2307/30037570 | MR 2167768 | Zbl 1159.11303
[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
[7] Ericksen, L.: Divisibility, iterated digit sums, primality tests. Math. Slovaca 59(3), 2009, p. 261–274 DOI 10.2478/s12175-009-0122-7 | MR 2505805 | Zbl 1209.11012
[8] Ericksen, L.: Iterated Digit Sums, Recursions and Primality. Acta Mathematica Universitatis Ostraviensis 14, 2006, p. 27–35 MR 2298910 | Zbl 1148.11007
[9] Ericksen, L.: Primality testing and prime constellations. Šiauliai Math. Semin. 3(11), 2008, p. 61–77 MR 2543450 | Zbl 1163.11007
[10] Harborth, H.: Über die teilbarkeit im Pascal-Dreieck. Math.-Phys Semesterber 22, 1975, p. 13–21 MR 0384676 | Zbl 0296.10009
[11] Harborth, H.: Ein primzahlkriterium nach Mann und Shanks. Arch. Math. 27(3), 1976, p. 290–294 DOI 10.1007/BF01224673 | MR 0417038 | Zbl 0325.10004
[12] Harborth, H.: Prime number criteria in Pascal’s triangle. J. London Math. Soc. (2) 16, 1977 p. 184–190 MR 0476623 | Zbl 0371.10002
[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 MR 0663657 | Zbl 0479.10005
[14] Kummer, E. E.: Über die Erganzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math. 44, 1852, p. 93–146
[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 MR 0306098 | Zbl 0239.10010
[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
[17] Robin, G.: Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann. J. Math. Pures Appl. 63, 1984, p. 187–213 MR 0774171 | Zbl 0516.10036
[18] Schinzel, A., Sierpinski, W.: Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4, 1958, p. 185–208, Erratum 5, 1959, p. 259 MR 0106202 | Zbl 0082.25802
[19] Weisstein, E.W.: Distinct Prime Factors.
[20] Weisstein, E.W.: Prime Constellation.
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