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Using six consecutive primes to create the points (59,61), (67,71), and (73,79) as vertices of a triangle with an area of just 2, the first prime of six is 59. Will the area ever again be 2? Can the area ever be an odd prime?

J. M. Bergot
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    If we take the least prime to be $2$, is not the area $3$? First example I tried. – lulu Jun 13 '19 at 18:13
  • What is the first prime of six consecutive ones that gave you a triangle with prime area 3? Did you find another triangle with area TWO? – J. M. Bergot Jun 13 '19 at 18:21
  • I believe I said, the least prime was $2$. I did not search systematically. You asked if the area could be an odd prime. Barring error, my example (again, the first I tried) proves it can. – lulu Jun 13 '19 at 18:25
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    Note that it is conjectured that "possible" prime combinations - ie those not barred by obvious congruence problems - so $p, p+2, p+8, p+12, p+14, p+20$ as here - will appear infinitely often, and this will simply translate the same triangle many times. (The test is that for any prime $p$ they don't cover every congruence class modulo $p$). – Mark Bennet Jun 13 '19 at 18:25
  • Should say: it's not obvious to me that the area is always an integer. It's clear that twice the area is an integer, but I don't immediately see why half integers are impossible. A quick search shows that the area is generally even (the reason for which is also not clear to me). – lulu Jun 13 '19 at 18:28
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    Starting with the prime $739$ gives area $2$. The vertices being $(739, 743),, (751 ,757),, (761, 769)$. – lulu Jun 13 '19 at 18:32
  • @lulu In the numerator of that formula, each term has a factor which is the difference of two odd primes. That makes the numerator even. – B. Goddard Jun 13 '19 at 18:34
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    I agree that first prime 2 will give area 3, but 2 is an even prime. What about if all six are odd primes--any areas that are also primes? If you think the area is not always an integer, then find examples. That the area is generally even gives you the reason why I posed the question. You stated that the prime gaps will be 2, 6, 4, 2, 6 and that they will occur infinitely often. Can there be a different collection of gaps to also give area 2? – J. M. Bergot Jun 13 '19 at 18:35
  • @B.Goddard Ah, good point. Thanks. – lulu Jun 13 '19 at 18:36
  • Our comments crossed. Note that your example has prime gaps of 4. 8, 6, 4, 8 which differs from the gaps I sent. – J. M. Bergot Jun 13 '19 at 18:43
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    The slopes of the sides given by the triangle with starting prime 739 differs from the slopes of the sides given by triangle with starting number 59. One wonders just how many differently shaped triangles will give areas of 2. – J. M. Bergot Jun 13 '19 at 19:00
  • It could happen that the area is a semiprime: divide by 2 to see how large the prime is. – J. M. Bergot Jun 13 '19 at 19:47
  • Early results at http://oeis.org/history?seq=A308649 – J. M. Bergot Jun 14 '19 at 19:09

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