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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
Summary:
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.
References:
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