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Let's say for a prime number $P$, I compute the sum involving $P' + P''$, where $P'$ is the largest sub-prime number below $P$ and $P''= P - P'$, such that $P''$ is the largest sub-prime below P' to fit in this sum exactly.

For example:

$97 = 89 + 5 +3$ (3 sub-primes),

$127 = 113 + 11 + 3$ (3 sub-primes),

$ 541 = 523 = 523 + 13 + 5$ (3 sub-primes),

$360,749 = 360,653 + 89 + 7$ (3 sub-primes),

$80,873,624,627,236,069 = 80,873,624,627,234,849 + 1117 + 3$ (3 sub-primes)

If it may exceed $4$, is there an upper limit to the number? Just inquisitive, my background isn't in mathematics. Thank you.

Kaustubh Sinha
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  • The Pillai sequence is relevant here. – lulu Jun 11 '21 at 16:18
  • @KeithBackman Why are we talking about composites when the question is only related to primes? P+1 will always be a composite.. – Kaustubh Sinha Jun 11 '21 at 16:38
  • My bad. I'll remove the comment because it is inapt. – Keith Backman Jun 11 '21 at 16:45
  • @lulu Thank you. It seems this is is most akin to a subset sum problem. I wonder if this could be a way to determine new primes by determining and utilizing sub-primes which occur most frequent. – Kaustubh Sinha Jun 11 '21 at 16:45
  • As I recall, it is not difficult to show that the Pillai sequence extends forever, but it grows preposterously quickly. And your sequence would be worse since you are restricting to primes (Pillai$[n]$ is the least integer such that the greedy algorithm yields at least $n$ steps, while yours requires the last prime). Makes it hard to imagine saying much. – lulu Jun 11 '21 at 18:55

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