Triangular Number Matrix and Application in Cryptography: A Novel Pattern-Based Approach

Purpose: The calculation of triangular numbers using the conventional formula $T_n=n\left(n+1\right)/2$ becomes computationally infeasible for astronomically large values of n (e.g., numbers with 10 trillion digits), due to limitations in modern computing systems. Existing methods struggle with the division and multiplication of extremely large integers, resulting in either failure or prohibitively slow processing times. This paper proposes a novel triangular number system that leverages mathematical patterns and constants to derive efficient computational shortcuts for determining high-digit triangular numbers. By circumventing the inefficiencies of traditional arithmetic operations, this approach aims to enable the rapid calculation of triangular numbers at scales previously deemed intractable. The proposed methodology synthesizes recursive pattern recognition and optimized calculation techniques to overcome the bottlenecks inherent in current computational frameworks. Experimental validation, benchmark comparisons, and theoretical analysis indicate that the system offers substantial advantages in handling ultra-large numbers, representing a significant advancement in computational number theory. The application of this pattern based analysis of triangular numbers has also been formulated and shown using TLV (Triangular layered verification) based cryptographic protocol.