Prime constellations in triangles with binomial coefficient congruences

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