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 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.

We quantized the full Einstein equations in a globally hyperbolic spacetime $N=N^{n+1}$, $n\ge 3$, and found solutions of the resulting hyperbolic equation in a fiber bundle $E$ which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second order ordinary differential equation in $\R[]_+$. In case $n\ge 17$ and provided the cosmological constant $\Lam$ is negative the temporal eigenfunctions are eigenfunctions of a self-adjoint operator $\hat H_0$ such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space.

In a recent paper we presented a model for a unified quantization of gravity with other fundamental sources of nature. After quantization we had to solve a Wheeler-DeWitt equation which was a hyperbolic equation in a fiber bundle which was equipped with a Lorentzian metric with a time function $t$ ranging from $0$ to infinity. The Lorentzian metric had a big bang singularity in $t=0$ and the coefficients of the hyperbolic operator also inherited this singularity. The solutions of the Wheeler-DeWitt equation were products of spatial and temporal eigenfunctions and in order to prove that these solutions also experience a big bang singularity in $t=0$ the temporal eigenfunctions $w(t)$ had to become unbounded if $t$ tends to $0$. In our former paper this was only proved in special cases where $w$ could be expressed with the help of Bessel functions but not in general. In the present paper we prove that also in the general case the temporal solutions become unbounded near $t=0$, or more precisely: $\lim_{t\ra 0} (\abs w^2+t^2 \abs{\dot w}^2)=\un$ and $\limsup_{t\ra 0}\abs w^2=\un$.

Let O_25 be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge 25 whose composition factors are the irreducible quotients of reducible Verma modules. We show that O_25 is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl_2-modules. We also show that this sl_2-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge 1. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge 25 as a PSL_2(C)-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of SL_2 at level -1, showing that it has a symmetric tensor category of representations equivalent to Rep PSL_2(C). This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges 1 and 25, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.