Problems & Puzzles: Puzzles

Puzzle 530. Twins pointing to equidistant trios

Let's suppose we have a pair of twins primes (p, p+2).

Now let's use them as indexes. This way we get the following two primes: prime(p) & prime(p+2) which implicitly suppose the inner prime prime(p+1), and thus a trio of primes.

¿How often the trio of primes thus formed are equidistant and how far they can be in the trio?

I have gotten the smallest examples for trio primes distances 6 & 12:

 p p+2 prm(p) prm(p+1) prm(p+2) x 107 109 587 593 599 6 239 241 1499 1511 1523 12 next? ?

Q. Can you send the smallest case for larger x-values?

Contributions came from Torbjörn Alm (x<=48), J-C Colin (x<=90), Antoine Verroken (x<=48), Farid Lian (x<=96), Farideh Firoozbakht (x<=102 & a proof), J.K. Andersen (x<=216)

***

Farideh wrote:

Please consider the following statement as my contribution for this puzzle.

" The smallest prime p for x = 6k, k = 1, 2, ..., 17 are respectively
107, 239, 51719, 90071, 82757, 792227, 1673279, 724781, 5571149, 12292391,
15251777, 21625061, 59676821, 60932999, 46985747, 105238871 & 218472767. "

...

She wrote about why x is a multiple of 6:

Yes, since x is even, we have three cases x=6k+2 , x=6k+4 & x=6k .

1. x=6k+2 : In this case we have two cases prm(p)=3k'+1 or prm(p)=3k'+2 .

If prm(p)=3k'+1 then prm(p+1)=3k'+1+x=3k'+1+6k+2=3(k'+2k+1) so 3 divides
prm(p+1) which is impossible.

If prm(p)=3k'+2 then prm(p+2)=3k'+2+2x=3k'+2+2(6k+2)=3(k'+4k+2) so 3 divides prm(p+2) which is impossible.

2. x=6k+4 : In this case we have two cases prm(p)=3k'+1 or prm(p)=3k'+2 .

If prm(p)=3k'+1 then prm(p+2)=3k'+1+2x=3k'+1+2(6k+4)=3(k'+4k+3) so 3 divides prm(p+2) which is impossible.

If prm(p)=3k'+2 then prm(p+1)=3k'+2+x=3k'+2+6k+4=3(k'+2k+2) so 3 divides
prm(p+1) which is impossible.

Hence x cannot be of the forms 6k+2 or 6k+4. So the only possible case is x=6k  and the proof is complete.

***

Andersen wrote:

 p prm(p) x 107 587 6 239 1499 12 51719 635003 18 90071 1160519 24 82757 1058891 30 792227 12067751 36 1673279 26833577 42 724781 10971743 48 5571149 96496979 54 12292391 223260823 60 15251777 280499551 66 21625061 405722197 72 59676821 1183743023 78 60932999 1210004753 84 46985747 920126959 90 105238871 2150518397 96 218472767 4632520727 102 2455300511 58295054821 108

 p prm(p) x 1498852841 34812231409 114 1826163461 42791999363 120 4169166767 101294810657 126 2609351399 62118833507 132 4672485107 114080304691 138 8201143019 205051764989 144 4144782197 100676987581 150 9849272429 248141843777 156 40985013239 1093465976429 162 16160665949 415496509471 168 59137642829 1600331947583 174 64250122721 1744222494143 180 63672555809 1727944666961 186 112083564491 3107608139359 192 155148609881 4354025197733 198 246682314419 7041548322169 204 206931015671 5869105501471 210 93971514269 2588225060021 216
The search is exhaustive for prm(p) < 2*10^13. The tables omit p+2, prm(p+1) = prm(p)+x, and prm(p+2) = prm(p)+2*x.

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