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The three points having consecutive prime co-ordinates (2,3), (11,13) and (47,53) are collinear. Are these the only three such points?

J. M. Bergot
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    (5,7) (11,13) (17,19) – Integrand Oct 05 '21 at 03:10
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    No, (29,31) doesn’t lie on the line joining the three given points since the slope joining (2,3) and (29,31) is $\frac{28}{27} $, which is not equal to $\frac{10}{9}$ . – Lai Oct 05 '21 at 03:25
  • Not true. Find the slopes (11-2) and (13-3) and (53-13) and (47-11) to get 10/9. With your (29,31) you get 28/27. – J. M. Bergot Oct 05 '21 at 03:31
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    The question is equivalent to prove that there are infinitely many consecutive prime pairs in the form (2+9n, 3+10n). – Lai Oct 05 '21 at 03:35
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    @bof I think OP meant a unique triple of prime coordinate collinear points. The set of three (without regard to order) is unique. – Deepak Oct 05 '21 at 03:45
  • Someone answered this by referring to Green-Tao theorem. What happened to the text sent in? You can vaporize the question I asked. – J. M. Bergot Oct 05 '21 at 05:51
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    @J.M.Bergot I deleted my answer because I failed to address the stipulation that the coordinates be consecutive primes. Would you like to relax that condition? In which case I will restore my answer. – Deepak Oct 05 '21 at 10:34
  • @Deepak Someone mentioned the Green-Tao theorem and that's my answer. Oddly, the text containing G-T has disappeared. – J. M. Bergot Oct 05 '21 at 23:42
  • @J.M.Bergot I see no answer by you (and I have the rep to see deleted answers). On the other hand I had an answer using G-T which I deleted for the reason I stated. I can restore my answer if you wish. – Deepak Oct 06 '21 at 03:17
  • @Deepak After the G-T theorem nothing further need be said, for it IS the answer to my question. – J. M. Bergot Oct 06 '21 at 05:23
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    @J.M.Bergot You asked about consecutive prime pair coordinates. G-T doesn't help with that (as far as I can figure). But I am undeleting my answer all the same. – Deepak Oct 06 '21 at 08:48
  • @Deepak I posed the question with the condition that the first prime point is the minimum, (2,3). By accident I found the other two points (11,13) and (47,53). If the original point were (11,13) and the middle point were (47,53) I think it would be difficult to find the larger third point being on the same line as the lesser two because finding larger consecutive primes for this last point would mean that the gap between them would be large. This is a puzzle you can work on: let the first two points be (11,13) and (47,53) and find the larger third point composed of consecutive primes. – J. M. Bergot Oct 06 '21 at 23:31

1 Answers1

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The number of such triples (or indeed any multiple) with collinear points with prime coordinates cannot be limited.

By the Green-Tao theorem, the sequence of prime numbers contains arbitrarily long arithmetic progressions. If we choose the first point to be a prime pair $(p, q)$ in such a sequence, then we can choose $(p+a, q+b)$ and $(p+2a,q+2b)$ (both being prime coordinate pairs) to follow on. The theorem guarantees their existence.

Note: as pointed out in the comments (and acknowledged by me), I am not able to address the condition of consecutive prime coordinates. However, as the OP feels that Green-Tao theorem is adequate to address his question, I am undeleting my answer.

Deepak
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  • Using that theorem, find examples of consecutive primes (p,q) that satisfy the addition of a and b and of 2a and 2b to have all three being collinear. It would make quite a sequence for minimum values for p and q. – J. M. Bergot Oct 05 '21 at 04:05
  • @Deepak Note your answer doesn't fully resolve the question as it's specifically asking for points having consecutive prime co-ordinates. – John Omielan Oct 05 '21 at 04:16
  • @JohnOmielan OK I see now. The pair of coordinates in a point comprise consecutive primes. I will delete. – Deepak Oct 05 '21 at 04:25
  • @Deepak No worries. The OP should have specifically stated consecutive prime pair co-ordinates. – John Omielan Oct 05 '21 at 04:26