On the Non-Existence of a Prime Quadruplet With Consecutive Gaps 8,6,8 and Unit Digit Sequence 9,7,3,1.

We investigate the existence of a prime quadruplet (𝑃1, 𝑃2,𝑃3,𝑃4 ) of consecutive primes greater than 3 whose consecutive successive gaps are 8,6,8 and whose base-10 units follow the sequence 9,7,3,1. Writing 𝑃2=𝑃1+8, 𝑃3=𝑃1+14, 𝑃4=𝑃1+22, the prescribed unit-digit pattern is fully compatible with these relations modulo 10. However, reducing the same expressions modulo 3 shows that for every possible residue class of 𝑃1 (mod 3), at least one of 𝑃1,𝑃2,𝑃3,𝑃4 becomes divisible by 3 and exceeds 3, contradicting primality. Thus, no such quadruplet exists. This establishes a deterministic structural impossibility-a “prime void” – arising from the interaction of specific gap templates and residue-class constraints. The argument is elementary, yet to the best of our knowledge this particular forbidden configuration has not appeared in the literature.