Someone asked me about sexy cousins. I'll admit, I was briefly hopeful. Turns out they meant primes with gap 6 and gap 4 respectively.

Disappointed, I did what anyone would do and stayed up computing distinct-prime partitions of primes.

Define m_d<(p) as the minimum k such that prime p is a sum of k pairwise distinct primes, each strictly less than p. Set m_d<(p) = ∞ if no such decomposition exists.

For every prime p ≤ 10^8 (verified directly, ~30 seconds on commodity hardware):

• m_d<(11) = ∞, uniquely. 11 simply refuses to be built from smaller primes. Provable by exhaustion: the 11 subsets of {2,3,5,7} of size ≥ 2
sum to {5,7,8,9,10,12,14,15,17}. 11 is not invited.

• m_d<(17) = 4, uniquely. 17 needed all four of its juniors (2+3+5+7) just to show up. Overachiever or socially awkward — you decide.

• m_d<(p) ∈ {2, 3} for every other prime in range. 5,761,448 tested. All well-adjusted.

I don't know if this object has a name. I don't know if this is trivially known, or trivially reducible to something known.

The repo has code, a writeup, verification prompts you can feed to any LLM to check the claims, and some other things I found along the way.

https://github.com/keeltremor/goldeen
https://doi.org/10.5281/zenodo.19542143

If any of this rings a bell please let me know!
🐉

#numbertheory #primes

#numbertheory #partitions #primes #sexyprimes