Bounded gaps and perfect power gaps in sequences of consecutive primes

We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also presented.

💡 Research Summary

The paper investigates three interrelated questions about gaps between consecutive primes: (i) whether several consecutive gaps can all be relatively large at the same time, (ii) whether all of them can be perfect squares or perfect powers, and (iii) whether it is possible to have long blocks of consecutive primes none of whose pairwise differences is a square. The author combines classical sieve methods, recent breakthroughs on bounded prime gaps (Goldston‑Pintz‑Yıldırım, Zhang, Maynard, the Polymath project) and combinatorial arguments to obtain a series of new results.

Theorem 1. For any integer (k\ge2) and sufficiently large (x) the quantity
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