A 'prime anomaly' I was chasing turned out not to be about primes at all

I initially believed that the intriguing result I had found in prime gaps would be evidence of a deeper property specific to prime numbers. However, upon conducting a falsifiable test on squarefree numbers, which are integers with no repeated prime factors, I discovered that the same pattern manifested there as well, but with even stronger correlations.

This finding disproved my initial hypothesis that the effect was unique to primes. The squarefree gaps demonstrated a large lag-1 z-score at base 2 that was nearly four times larger than the corresponding z-score for prime gaps. The reason behind this phenomenon lies in the bounded nature of squarefree gaps, which are limited to a maximum value of 10.

Consequently, for bases greater than or equal to 11, the sum of digits of a squarefree gap is directly equivalent to the gap value itself. This accounts for the base-specific patterns observed in the squarefree gaps. Although the squarefree gaps do exhibit serial memory, this does not imply a deeper connection between the two types of gaps.

Instead, it serves to invalidate the notion that prime gaps possess any unique or fundamental properties. Moving forward, I plan to explore whether the same effect persists when applied to a thinned sequence with unbounded gaps of comparable density. The natural control for this experiment is the sums of two squares, which have a similar density but allow for gaps that grow without bound.

By conducting this follow-up study, I aim to shed further light on the underlying structure of prime and squarefree gaps.