[seqfan] Where are the primes?

[David W. Wilson]

Richard Guy wrote:

> You can get an earlier AP of 12 primes if you don't insist on the minimum
> common difference.   E.g.,  a = 23143,  d = 30030  [V.A. Golubev, probably
> one of the references in  A5  in  UPINT.]       R.

My computations were in regard to finding an AP with the theoretically
smallest possible difference.  If the Prime Patterns conjecture is true, then
an upper bound on this minimal difference is n# (n primorial) where n is
the number of elements in the AP.  Prior to my search, a 12-element AP
of primes theoretically had minimal difference 2310, whereas now this is
established fact.

In brief, if the difference of an AP of n primes is less than n#, then some
prime p <= n does not divide this difference, and therefore divides one of
the elements.  This element, being prime, must be p itself.  Therefore, an
AP of primes either has difference >= n# or contains a prime element
<= n.

The latter case includes a finite number of AP's which can at worst be
exhaustively checked (but we can streamline the search).  This search
can produce a AP of primes of difference < n#, as with the AP of 7 primes
starting at 7 of difference 150.

If there is no AP with difference < n#, then we consider an AP of n
elements with difference n#.  Such an AP does not cover all the residues
of any prime (for p <= n, all elements will all have the same residue,
since p divides n#, for p > n, the n elements cannot possibly cover all p
residues).  These are precisely the conditions under which the Prime
Patterns Conjecture asserts an infinitude of prime patterns in the
form of a n-element AP of difference n#.

In his article "Prime Arithmetic Sequence", Weisstein originally had
an incomplete sequence of theoretical minimum differences for a
prime AP of n elements.  As an early element was missing, Sloane did
not have the sequence (? is not a valid element in a Sloane sequence).
Based upon the above argument assuming PPC, I filled in the gaps,
and extended the sequence to Sloane's A033188.

Finding a starter element that confirms a given minimum difference
is, of course, rather another matter.  I easily computed a starters for
n <= 10, and so began A033189.  Weisstein then blindsided me by
finding a(11), but I fought back with a(12) and a(13).  The idea of
A033189 is to find the smallest starter element that confirms the
corresponding difference in A033188.