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Number Theory & Arithmetic LycéeExplication9 min

The Middle of Two Neighbouring Primes Is Never Prime

The midpoint of p and q sits strictly between them, and consecutive means precisely that no prime lives in that interval, so the answer is never and the proof is two lines with no arithmetic in it. The pair 2 and 3 survives for a different reason, since five halves is not an integer, and it is the only such pair.