I took number $3$ and observed:

$3$ is an arithmetic progression of length one.

$3,5$ is an arithmetic progression of length two.

$3,5,7$ is an arithmetic progression of length three.

Then I took number $5$ and observed:

$5$ is an arithmetic progression of length one.

$5,7$ is an arithmetic progression of length two.

$5,11,17$ is an arithmetic progression of length three.

$5,11,17,23$ is an arithmetic progression of length four.

$5,11,17,23,29$ is an arithmetic progression of length five.

So surely it would be nice to know:

Is it true that for every prime number $p$ there exists $p$ arithmetic progressions, first of length one, second of length two, ... , $p$-th of length $p$ so that every of those arithmetic progressions has number $p$ as its first term.

I know about Green-Tao theorem but I do not know does this can follow from it or can the combination of Green-Tao theorem with some proven or unproven facts answer this question?

It is clear that with $p$ as a starting point we cannot have an arithmetic progression of primes with $p+1$ terms because $(p+1)$-st term would be $p+pd$ which is composite.

If this question has affirmative answer then we would have that for every natural $k$ we have an infinite number of arithmetic progressions of primes with $k$ terms so I guess that this is extremely hard but would like to hear opinions and suggestions for how could this be attacked.

  • 3
    Note that all you need is a single one of length $p$. The others could be initial parts of it. – almagest Jun 18 '16 at 16:40
  • 1
    But this conjecture looks way beyond the limits of current knowledge to me. It is far stronger than the Green-Tao result. – almagest Jun 18 '16 at 16:46
  • 1
    For $p=7$ you have $7,157,307,457,607,757,907$. – Ángel Valencia Jun 18 '16 at 16:53
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    @Farewell Another slightly better sequence, we have reached 10-length, one step below the goal: $11 ,$ $224494631 ,$ $448989251 ,$ $673483871 ,$ $897978491 ,$ $1122473111 ,$ $1346967731 ,$ $1571462351 ,$ $1795956971 ,$ $2020451591$ – Joffan Jun 18 '16 at 18:07
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    Arithemtic primes and records: primerecords.dk/aprecords.htm#minimalstart – Michael Stocker Jun 19 '16 at 7:27

Collecting my results together, there is a $7$-length sequence found for $7$:

$7,$ $157,$ $307,$ $457,$ $607,$ $757,$ $907$

Note that, starting at $11$ (or larger prime), to have a sequence of length $7$ or more it is essential that the step is divisible by $7\#=210$, to avoid multiples of the lesser primes, and in general for the $p$-length sequence from $p$, the step has to be divisible by the primorial of the previous prime to $p$ (which could be written $(p-1)\#$, since only primes are multiplied anyway).

Best sequences to date for first 20 primes: $$\tiny \begin{array}{c|cc} \text{Length} & & &\text{Longest sequence}\\ \hline 2 \;\checkmark & 2 & 3 & & & & & & & & \\ 3 \;\checkmark & 3 & 5 & 7 & & & & & & & \\ 5 \;\checkmark & 5 & 11 & 17 & 23 & 29 & & & & & \\ 7 \;\checkmark & 7 & 157 & 307 & 457 & 607 & 757 & 907 & & & \\ 10 & 11 & 224494631 & 448989251 & 673483871 & 897978491 & 1122473111 & 1346967731 & 1571462351 & 1795956971 & 2020451591 \\ 10 & 13 & 111739753 & 223479493 & 335219233 & 446958973 & 558698713 & 670438453 & 782178193 & 893917933 & 1005657673 \\ 9 & 17 & 6947 & 13877 & 20807 & 27737 & 34667 & 41597 & 48527 & 55457 & \\ 10 & 19 & 35707369 & 71414719 & 107122069 & 142829419 & 178536769 & 214244119 & 249951469 & 285658819 & 321366169 \\ 9 & 23 & 100613 & 201203 & 301793 & 402383 & 502973 & 603563 & 704153 & 804743 & \\ 9 & 29 & 8456519 & 16913009 & 25369499 & 33825989 & 42282479 & 50738969 & 59195459 & 67651949 & \\ 10 & 31 & 104816281 & 209632531 & 314448781 & 419265031 & 524081281 & 628897531 & 733713781 & 838530031 & 943346281 \\ 10 & 37 & 2040607 & 4081177 & 6121747 & 8162317 & 10202887 & 12243457 & 14284027 & 16324597 & 18365167 \\ 9 & 41 & 19489511 & 38978981 & 58468451 & 77957921 & 97447391 & 116936861 & 136426331 & 155915801 & \\ 9 & 43 & 52963 & 105883 & 158803 & 211723 & 264643 & 317563 & 370483 & 423403 & \\ 9 & 47 & 3025307 & 6050567 & 9075827 & 12101087 & 15126347 & 18151607 & 21176867 & 24202127 & \\ 9 & 53 & 15441983 & 30883913 & 46325843 & 61767773 & 77209703 & 92651633 & 108093563 & 123535493 & \\ 9 & 59 & 49267739 & 98535419 & 147803099 & 197070779 & 246338459 & 295606139 & 344873819 & 394141499 & \\ 9 & 61 & 27388681 & 54777301 & 82165921 & 109554541 & 136943161 & 164331781 & 191720401 & 219109021 & \\ 9 & 67 & 48175117 & 96350167 & 144525217 & 192700267 & 240875317 & 289050367 & 337225417 & 385400467 & \\ 9 & 71 & 13959401 & 27918731 & 41878061 & 55837391 & 69796721 & 83756051 & 97715381 & 111674711 & \\ \end{array} $$

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