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This is about prime numbers such that at least three points $(i,p_i)$, $(j,p_j)$ and $(k,p_k)$ are on the same straight line.

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Conjectures:

  1. For any pair $(i,p_i),\, i>1$, there are two different primes $p_j,p_k>p_i$ such that $(i,p_i)$, $(j,p_j)$ and $(k,p_k)$ are on a straight line. Tested for the first 25,000 primes.
  2. Any pair $(i,p_i), \,i>1$, is co-linear with at least two such straight lines. Tested for the first 200 primes. The sets below contain a maximal number of prime points. $\{(2,3),(3,5),(4,7)\},\{(4,7),(5,11),(17,59),(19,67),(20,71),(22,79),(23,83)\},\{(6,13),(7,17),(10,29),(12,37),(13,41),(16,53),(18,61),(21,73)\},\{(2,3),(5,7),(8,19)\},\{(4,7),(6,13),(8,19)\},\{(3,5),(5,11),(7,17),(9,23)\},\{(8,19),(9,23),(11,31),(14,43),(15,47)\},\dots$
  3. Given an integer $n>0$ there are $n$ different primes $p_{k_1},\dots ,p_{k_n}$ such that $(k_1,p_{k_1}),\dots,(k_n,p_{k_n})$ are on a straight line. In the table: for $1<i<j<l\in\mathbb N$ there are $m$ straight lines thru $(i,p_i),(j,p_j)$ and $n$ is the record of number of points $(h,p_h)$ on a line.

$$ \begin{array}{rrr} l & m & n \\ \hline 5 & 1 & 3 \\ 10 & 7 & 8 \\ 20 & 27 & 8 \\ 50 & 148 & 20 \\ 100 & 400 & 20 \\ 200 & 1338 & 20 \\ 500 & 6253 & 31 \\ 1000 & 16859 & 31 \\ 2000 & 49460 & 52 \end{array} $$

My question is if there is something known about such co-linear primes?

Lehs
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  • I assume that when you recite $p_i$, you mean an arbitrarily indexed prime, not the $i$th prime. I don't think you mean to imply that $p_{27}, p_{28}, p_{29}$ (or any other three sequential prime) must fall on a line. Please clarify. – Keith Backman Jul 13 '22 at 15:26
  • @KeithBackman - $p_i$ do certainly means the i-th prime, but there are only three consecutive primes that makes a line: 3,5,7. – Lehs Jul 13 '22 at 15:59
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    @Lehs consecutive primes will be collinear if and only if they form an arithmetic progression. It's believed that there exist arbitrarily many consecutive primes in arithmetic progression (this is stronger than the Green-Tao theorem but weaker than the first Hardy-Littlewood conjecture); if so, this would imply your third conjecture. But the current record is only 10 consecutive primes in arithmetic progression, so evidently it's easier to find collinear primes than consecutive collinear primes. – Ravi Fernando Jul 14 '22 at 16:15
  • @RaviFernando - Thanks! I will look this up. – Lehs Jul 14 '22 at 17:20

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