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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...

We examine the congruences and iterate the digit sums of integer sequences. We generate recursive number sequences from triple and quintuple product identities. And we use second order recursions to determine the primality of special number systems.

The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials $a(n;x)$ defined by $a(0;x)=0$, $a(1;x)=1$, $a(2n;x)=a(n;x^2)$, and $a(2n+1;x)=xa(n;x^2)+a(n+1;x^2)$; these polynomials are Newman polynomials, i.e., they have only 0 and 1 as coefficients. In this paper we prove numerous reducibility and irreducibility properties of these polynomials, and we show that cyclotomic polynomials play an important role as factors. We also prove several related results, such as the fact that $a(n;x)$ can only have simple zeros, and we state a...