The Prime Density Triangle offers a novel combinatorial framework for exploring properties of prime numbers. From the algebraic relations embedded within this triangle, specifically the coupled expressions X^p1± p1and X^p0± p0, we derive two closed-form sequences.First, the multiplication of these expressions yields the sequence Sn(m) = (mn+ n2)1/(n+2), which converges to m. Second, by a deliberate construction—replacing the additive denominator n + 2 with the multiplicative denominator 2n—we obtain the Exponential-Polynomial Root Mean (EPRM) defined by Rn(m) = (mn+ n2)1/(2n) for m > 0 and n ∈ N. This construction provides a closed-form, non-iterative method for approximating √m for sufficiently large n. In this paper, we present a complete derivation of both sequences from the Prime Den- sity Triangle’s algebraic frameworks, rigorous convergence proofs including asymptotic error analysis, detailed worked examples for m = 2, 3, 5 with explicit computations, numerical summary tables, explicit iterations for classical methods (Babylonian and Newton-Raphson), and a comprehensive comparison. The approach is elementary and highlights the dominance of exponential terms over polynomial corrections.

We introduce a triangular array of integers defined by a(n, k) = k(k + 1)n−k for n ≥ 1 and 1 ≤ k ≤ n, which we call the researcher’s triangle. The construction is multiplicative and differs from classical additive triangles such as Pascal’s. We give combinatorial interpretations using a leaders-followers model and a rooted hierarchical structure model. We identify several diagonal sequences as classical figurate numbers, including powers of two (A000079), natural numbers (A000027), and oblong numbers (A002378), and establish connections to triangular and tetrahedral numbers. We derive a bivariate generating function and a recurrence relation, and we discuss the row sums.

We introduce the prime density triangle, a multiplicative array constructed by a modified rule of indices. Two display formats are given: a right-angle triangle with converted and unconverted entries, and an equilateral triangle obtained by mirroring the right-angle triangle about the central term xn. We derive the row-sum generat- ing function and provide worked examples using both direct computation and the generating function. Using a four-rule division scheme applied to the unconverted rows we obtain the sequence whose integer parts sum to the magic numbers for two spin orientations. Replacing the repeated end integers by 1,1 yields integer parts that give the standard harmonic oscillator (HO) magic numbers and, in conjunction with the two-spin-orientation nucleon magic numbers, gives precisely the standard nuclear magic numbers 2, 8, 20, 28, 50, 82, 126, 184. Furthermore, digital-root reduc- tion of the converted triangle reveals a distinct “4,9,9” pattern whose diagonals concatenate into large primes, establishing a direct arithmetic link between prime distribution and the symmetry conditions of nuclear shell closures. Subshell filling up to x9, deductions, and evidence from nuclear physics are included. A recurrence M = m ± [(n1n2) + 2] is presented for generating further magic numbers.The sequences appear as OEIS A005897, OEIS A018226 and OEIS A007290.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. P. Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). We present the problems, conjectures related to this problems and we summarize the know results. We make the following conjecture: \par \noindent{\bf Conjecture } $$\lim\sb {n \to \infty} {f(n)-n \over \sqrt n} = \sqrt {2 + \frac{4}{9}}.$$

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on $4$ vertices. We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$.

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \bigcup P_3$ but not contains a cycle on $4$ vertices and not contains $Z_4$.

Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-H, n)$ for $n\geq 4r+10, r\geq 3, r+1 \geq k \geq 4$ where $H$ is a graph on $k$ vertices which contains a tree on $4$ vertices but not contains a cycle on $3$ vertices. We also determine the values of $\sigma (K_{r+1}-P_2, n)$ for $n\geq 4r+8, r\geq 3$.

"A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $\sigma(K_4-e,n)$ denote the smallest degree sum such that every $n$ -term graphical sequence $S$ with $\sigma(S)\geq\sigma(K_4-e,n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma(K_4-e,n)$ . In this paper, we prove that $\sigma(K_4-e,n)=2[(3n-1)/2]$ for $n\geq7$ and $n=4,5$ , and $\sigma(K_4-e,6)=20$ .''

"In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$ -term graphical sequence, and $\sigma(S)$ be the sum of the terms in $S$ . Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$ -term graphical sequence $S$ having $\sigma(S)\geq l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $\sigma(H,n)$ . We show $\sigma(C_{2m+1},n)=m(2n-m-1)+2$ , for $m\geq 3$ , $n\geq 3m$ ; $\sigma(C_{2m+2},n)=m(2n-m-1)+4$ , for $m\geq 3$ , $n\geq 5m-2$ .''

In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.