Arithmetic progressions of primes and sums of squares

[Hugo Pfoertner]

Don Reble wrote:
> 
> Seqfans, Dr. Guy:
> 
> > I wrote a Fortran program...
>         You poor fellow. :-(

I have written Fortran for more than 30 years and I can _think_ Fortran.
- and it's still hard to beat the speed of a well written Fortran
program together with a state-of-the art compiler.

> 
> > My a(7) seems to be correct, with the progression
> > 7,157,307,457,607,757,907.
>     Yes.
>     That 1307 is an easy mistake to make. If one mistakenly believes
>     that the gap must be >= 2*3*5*7, he won't find your progression.
> 
>     Dr Guy's UPiNT2 A5 is interesting. He has A005115 there, with
>     l(7)=1307, even though the next paragraph refutes it:
>     "(q,d)=(7,150)". Alas, he has just sent UPiNT3 to the publisher;
>     no doubt he will fix UPiNT4.
> 
> > Currently I'm searching for a(15)...
>     If I may steal your thunder,
>     a(15)=173471351, gap(15)=4144140

The thunder is still there! My program found a smaller a(15) this
evening:
 a(15)=136023703, gap(15)=6936930 (please check!)

>     a(16)=198793279, gap(16)=9699690 (This one's in UPiNT2 A5.)

The program will search up to 400*10^6, so I will get a result for a(16)
during the next few days.

> 
>     Dr Guy doesn't claim that the progressions of A5 have the lowest
>     possible ending, so the other ones are only upper-bounds of A005115.
> --
> Don Reble       djr at nk.ca

Some additional info: In my previous message the correct reference
should be Green & Tao, not Terence & Tao.

The next term A006560(7) (7 consecutive primes in arithmetic
progression) might be in
(from Chris Caldwell's prime pages: Arithmetic Progressions of Primes, 
http://primes.utm.edu/top20/page.php?id=14 )
<<H. Dubner and H. Nelson, "Seven consecutive primes in arithmetic
progression," Math. Comp., 66 (1997) 1743-1749.  MR 98a:11122 
Abstract: It is conjectured that there exist arbitrarily long sequences
of consecutive primes in arithmetic progression. In 1967, the first such
sequence of 6 consecutive primes in arithmetic progression was found.
Searching for 7 consecutive primes in arithmetic progression is
difficult because it is necessary that a prescribed set of at least 1254
numbers between the first and last prime all be composite. This article
describes the search theory and methods, and lists the only known
example of 7 consecutive primes in arithmetic progression. >>

Is it known if Dubner and Nelson found the smallest occurrence?

Hugo