On the Threshold Even Number for Universal Prime Summations

This paper explores the behavior of even numbers with respect to their representation as the sum of multiple prime numbers.While the well-known Goldbach Conjecture asserts that every even number greater than two can be written as the sum of two primes,this study investigates the point beyond which every even number can also be expressed as the sum of multiple prime numbers.We introduce the concept of a threshold even number EthresholdE_ such that for all E>Ethr > E_{threshold}E>Ethreshold,there exist valid representations using primes for all k∈[2,10]k \in [2, 10]k∈[2,10].Using combinatorial generation of partitions and prime filtering, we estimate that this threshold lies at E=36E = 36E=36,beyond which multi-prime representation becomes saturated for all k in the range considered.