Arithmetic progressions of primes and sums of squares

[Pfoertner, Hugo]

SeqFans,

some weeks ago Richard Guy mentioned the topic of arithmetic progressions of
primes and the article by Ben Green and Terence Tao
"THE PRIMES CONTAIN ARBITRARILY LONG ARITHMETIC PROGRESSIONS", which is
available at
http://arxiv.org/abs/math/0404188

See also the announcement:

http://mathforum.org/epigone/sci.math.research/byrtrersnerd

I tried fo find related sequences in the OEIS and found (among some others):

%I A006560 M0927
%S A006560 2,3,251,9843019,121174811
%N A006560 n consecutive primes in arithmetic progression.
%H A006560 <a
href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html#primes_AP">
Index entries for sequences r\
  elated to primes in arithmetic progressions</a>
%K A006560 nonn
%O A006560 2,1
%A A006560 njas

I tried to extend A006560, but couldn't find more terms in the range up to
2^31 (2.147*10^9). Could someone with more computer power try to extend
A006560?


%I A005115 M0854
%S A005115 1,2,3,7,23,29,157,1307,1669,1879,2089,249037,262897,725663
%N A005115 Least prime which is the end of an arithmetic progression of n
primes.
%D A005115 R. K. Guy, Unsolved Problems in Number Theory, A5.
%H A005115 <a
href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html#primes_AP">
Index entries for sequences r\
  elated to primes in arithmetic progressions</a>
%e A005115 a(11)=249037 since 110437,124297,...,235177,249037 is an
arithmetic progression of 11 primes ending with 249\
  037 and it is the least number with this property. - Michael Somos Mar 14
2004
%K A005115 nonn
%O A005115 0,2
%A A005115 njas
%E A005115 a(11)-a(13) from Michael Somos, Mar 14 2004.

I wrote a Fortran program to further extend this sequence and was a little
surprised when I got the following results:

Length of    Gap     End of
progression        progression
  2           1           3
  3           2           7
  4           6          23
  5           6          29
  6          30         157
  7         150         907  different from A005115(7)
  8         210        1669
  9         210        1879
 10         210        2089
 11       13860      249037
 12       13860      262897
 13       60060      725663
 14      420420    36850999  new.

Currently I'm searching for a(15), the search running at 80*10^6.
My a(7) seems to be correct, with the progression 7,157,307,457,607,757,907.
In my opinion a(0)=1 should be removed, because most people don't consider
"1" to be a prime, and if "1" where a prime, then a(2) would  be 2,
representing the "prime" sequence (1,2) and the length 4 progression would
be (1,3,5,7).
The sequence should also be marked as "hard", and have a link to the T&T
paper.

Is anybody aware of a source for more terms? T&T say:
At the time of writing the longest known arithmetic progression of primes
was that found in 1993 by Moran,Pritchard and Thyssen
[28], who found that
11410337850553 +4609098694200 k
is prime for k =0 ,1 ,...,21.
In 2003,Markus Frind found the rather larger example 376859931192959
+18549279769020 k
of the same length.
[28] A.Moran,P.Pritchard and A.Thyssen, Twenty-two primes in arithmetic
progression, Math.
Comp.64 (1995),no.211,1337--1339.

The corresponding "gap" sequence (0?) 1 2 6 6 30 150 210 210 210 13860 13860
60060 420420 might also be a candidate for the OEIS.

On page 43 the Terence & Tao paper says:
"Applying Theorem 1.2 to the set of primes p=1(mod 4), we obtain the
previously
unknown fact that there are arbitrarily long progressions consisting of
numbers which
are the sum of two squares."

This enticed me into calculating a sequence similar to A005115:
Least number which is the end of an arithmetic progression of n numbers that
are the sums of two non-zero squares.
(arithmetic progressions in A000404)

     Length         Gap         End
          2           3           5
          3           3           8
          4           8          26
          5           8          34
          6          12          65
          7          24         146
          8          24         170
          9          24         194
         10          24         218
         11          24         242
         12          84        1445
         13          84        2225
         14          84        2309
         15          84        2393
         16          84        2477
         17          84        2561
         18          84        2645
         19          84        2729
         20          84        2813
         21          84        2897
         22        2772       71633
         23       21252      479581
         24       21252      664445
         25       21252      685697
         26       42504     1141625
         27       42504     1184129

a(28)>2.8*10^6

Example: a(6)=65: 5=2^2+1^2, 17=4^2+1^2, 29=5^2+2^2, 41=5^2+4^2, 53=7^2+2^2,
65=7^2+4^2

Any comments?

Hugo