Characterizations of prime \(k\)-tuples using binomial expressions (Q2908090)

Although the twin primes characterization in terms of binomial coefficients has already been given by \textit{K. Dilcher} and \textit{K. B. Stolarsky} [Am. Math. Mon. 112, No. 8, 673--681 (2005; Zbl 1159.11303)], the argument supplied in this paper to prove it is quite different. Instead of using the generating functions, here the author employs the following recursive formula for binomial coefficients NEWLINE\[NEWLINE{{n+\frac{p-3}{2}}\choose {p-2}} = {{n+\frac{p-1}{2}}\choose {p-1}} - {{n+\frac{p-3}{2}}\choose {p-1}} >0 NEWLINE\]NEWLINE applied for the primes both \(2n-1\) and also \(2n+1\).NEWLINENEWLINEBeyond such remarkable achievement, this short paper presents even new necessary and sufficient conditions for higher order tuples in terms of binomial coefficients, based on the characterization of prime numbers by \textit{J. P. D'Angelo} [J. Geom. Anal. 14, No. 2, 215--229 (2004; Zbl 1082.32024)].NEWLINENEWLINEFurther interesting characterizations for \(k\)-tuple primes have been established in another work still by \textit{Z. R. Königsberg} [Neural Parallel Sci. Comput. 19, No. 3--4, 323--330 (2011; Zbl 1247.11024)].