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.

This paper investigates the sum of all non-Fibonacci integers that lie strictly between consecutive Fibonacci numbers. Starting from the operation a(n) = Mn− Fn, where Mn is that sum and Fn the preceding Fibonacci number, we obtain a sequence whose first three terms are themselves Fibonacci numbers: 1, 8, 34. We extend the exploration to seven other arithmetic operations (subtracting the next Fibonacci, adding either, multiplying either, and dividing either), and we systematically search for Fibonacci numbers that appear in each resulting sequence. We term such Fibonacci entries “lucky Fibonacci numbers”. Among all variants, only the addition and subtraction operations yield any lucky Fibonacci numbers: 1, 8, 34 (subtract previous), 5 (subtract next, ignoring the negative initial term), 21, 55 (add next), and none for add previous, nor for multiplication or division. We then apply the same gap-summing idea to other recurrent sequences: Lucas, Pell, and Tribonacci. For each, we find a small set of lucky Fibonacci numbers. All conjectures are verified computationally up to n = 200 (and up to Fibonacci F400), with no further hits found. The growth rates are exponential, making further coincidences increasingly improbable.

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.

This note presents two original and creative divisibility tests for the integers 2 and 5. Both rules use the factorial of the last digit of a number, combined with a small correction when the factorial value would disrupt the required modular congruence. The resulting tests are mathematically equivalent to the classical last-digit rules but reach the same conclusion through an unconventional factorial-based approach. Each theorem includes a short proof sketch, an explanation of the mechanism, and four worked examples. The methods are valid precisely because 2 and 5 divide the base 10, allowing the last digit to fully determine the remainder modulo of these divisors. The work is intended as an accessible piece of recreational number theory for educational purposes.

In this article, we define a “ recursive local” algorithm in order to construct two reccurent numerical sequences of positive prime numbers (𝑈2𝑛) and (𝑉2𝑛), ((𝑈2𝑛) function of (𝑉2𝑛)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (𝑊2𝑛) defined for any integer n≥ 3 by : 𝑊2𝑛 = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. . In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 . Thus, thanks to this algorithm of “ascent and descent”, we can validate the strong Euler-Goldbach conjecture.