[seqfan] Re: Prime Sums In A Grid

[zak seidov]

See:

http://zak08.livejournal.com/5453.html

with example of solution,

zak


--- On Tue, 2/24/09, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:

> From: Maximilian Hasler <maximilian.hasler at gmail.com>
> Subject: [seqfan] Re: Prime Sums In A Grid
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Tuesday, February 24, 2009, 11:50 AM
> > From: David Wilson <davidwwilson at comcast.net>
> > So, can you fill an infinite grid with distinct
> integers so that any
> > two adjacent integers sum to a prime?
> On Mon, Feb 23, 2009 at 10:50 PM, 
> <franktaw at netscape.net> wrote:
> > Almost certainly.  Start with
> > 1 2
> > 4 (...)
> 
> Is it reasonable to conjecture that the corresponding
> sequence (say,
> filling this square by antidiagonals in the greedy way) is
> a
> permutation of the positive integers?
> 
> Maximilian
> 
> [1],
> [2, 4],
> [3, 9, 7],
> [8, 10, 22, 6],
> [5, 21, 19, 25, 11],
> [12, 26, 40, 18, 36, 20],
> [17, 35, 27, 13, 23, 47, 33],
> [14, 24, 32, 16, 30, 50, 56, 28],
> [15, 29, 65, 51, 31, 53, 57, 45, 39],
> [38, 44, 42, 62, 52, 48, 74, 82, 34, 58],
> [41, 59, 95, 89, 75, 49, 83, 99, 55, 69, 43],
> [60, 68, 54, 78, 92, 64, 90, 80, 94, 112, 70, 46],
> [37, 71, 113, 73, 101, 87, 67, 77, 117, 79, 61, 81, 63],
> [66, 102, 86, 84, 126, 110, 106, 72, 116, 154, 88, 76, 100,
> 104],
> [85, 91, 125, 107, 97, 131, 123, 121, 155, 153, 103, 105,
> 151, 93, 119],
> 
> Least unused integer so far: 96
> 
> 
> /* prime sum grid : fill infinite square grid by
> antidiagonals such
> that sum of 2 neighbours (vertical and horizontal) is
> always prime */
> psg( N=15, last_diag=[1], min_not_used=2, others_used=[])={
> local(new_diag);
> print(last_diag",");
> for( d=1+#last_diag,N /* fill the d-th antidiagonal */,
>   #new_diag < d && new_diag=vector( d );
>   for( j=1,d,
>     ! new_diag[ j ] && new_diag[ j ] = min_not_used
> + (d-min_not_used)%2;
>     while( setsearch( others_used, new_diag[j] )
>       | ( j > 1 && setsearch( Set( vecextract(
> new_diag, 2^(j-1)-1)),
> new_diag[j] ))
>       | ( j < d && ! isprime( last_diag[ j ] +
> new_diag[ j ] ))
>       | ( j > 1 && ! isprime( last_diag[ j-1 ] +
> new_diag[ j ] ))
>     , new_diag[j] += 2;
>     );
>   );
>   others_used = setunion( others_used, new_diag );
>   while( setsearch( others_used, min_not_used )
>   , others_used = setminus( others_used, Set( min_not_used
> )); min_not_used++;
>   );
>   print( last_diag=new_diag"," ); /* a=concat(a,
> new_diag ); */
>  ); [ min_not_used, others_used ]; }
> 
> 
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