Let k >= 7 be fixed, and v = (55k + 556)/(26k + 672),1 < d < c < v, 1 < mu < nu < k(1-d/c) and mu <= N-2/N-1(d/c) <= v. Suppose that N '(1), N '(2) > 0 are real numbers, and that N-1, N-2 are real numbers satisfying N-1 > N '(1) and N-2 > N '(2). Then we prove the system of two Diophantine inequalities |p(1)(c) + . . . + p(k)(c )- N-1| < N-1(-(1/c)(v-c)) log(195) N-1, |p(1)(d) + . . . + p(k)(d) - N-2 | < N(2)(-(1/d)(v-d) )log(195) N-2 has prime solutions p(1), . . . ,p(k).
In Chapter 3 of his second notebook, Ramanujan defined numbers a (n, k) such that a (2, 0) = 1 and for n >= 2, a (n + 1, k) = (n - 1) a (n, k - 1) + (2n - 1 - k) a (n, k), where a(n, k) = 0 when k< 0 or k>n - 2. These numbers are expressed in terms of Stirling numbers of the first kind and associated Stirling numbers of the second kind, and satisfy certain divisibility properties. In this paper, we obtain further properties for a(n, k), including a new characterization of prime numbers. We also derive several congruences for the numbers of Ramanujan modulo p(2), some of which lead to new conditions for a prime number to be a Wilson prime.
This paper establishes explicit evaluations of the 2k-th power mean for generalized cubic Gauss sums. By exploiting analytic techniques and fundamental properties of classical Gauss sums, we derive closed-form expressions for these means. Furthermore, we develop a computationally efficient framework for analyzing higher-order moments of such sums.
In this paper, we consider one of graph invariants, called a general position number, of scale-free graphs, where the scale-free property is one of key properties of real-world networks. To understand the tendency of general position numbers of scale-free graphs, we primarily deal with deterministic artificial scale-free graphs, since such graphs are very manageable and artificial models of complex networks mimic many properties of real-world networks. We derive estimates for the general position number of three well-known hierarchical scale-free graphs generated by deterministic models: one model produces fractal graphs, while the other two generate non-fractal graphs.
