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# Generalized arithmetic progressions

A generalized arithmetic progression (GAP) (multiple arithmetic progression, $k\,$ -dimensional arithmetic progression) is defined as

$a(n_{1},\,\ldots ,\,n_{k})=a_{0}+\sum _{i=1}^{k}a_{i}\,n_{i},\quad 0\leq n_{i}\leq N_{i},\,$ where the $a_{i},\,i\,\geq \,0,\,$ are fixed.

The number $k\,$ , that is the number of permissible differences, is called the dimension of the generalized progression; arithmetic progressions (AP) being of dimension 1.

## Generalized arithmetic progressions (GAP) of primes

### 2 X n generalized arithmetic progressions (GAP) of primes

Here is the beginning of Granville's table: (Cf. A113830, A113831)

$n\,$ GAP Last term
2 3+8i+2j 13
3 7+24i+6j 43
4 5+36i+6j 59
5 11+96i+30j 227
6 11+42i+60j 353
7 47+132i+210j 1439

First example:

$3+8i+2j\,$ j \ i 0 1
0 3 11
1 5 13

### First known 3 X 3 X 3 generalized arithmetic progression (GAP) of primes

A 3 X 3 X 3 generalized arithmetic progression (GAP) of 27 primes, with 929 as its smallest prime and 27917 as its largest, given by

$929+2904i+3150j+7440k,\quad 0\leq i\leq 2,\,0\leq j\leq 2,\,0\leq k\leq 2.\,$ Mathematics majors Jeffrey P. Vanasse and Michael E. Guenette, working under the direction of Marcus Jaiclin and Julian F. Fleron of Westfield State College in Massachusetts, made the discovery. (Cf. A153412)

$k\,=\,0\,$ j \ i 0 1 2
0 929 3833 6737
1 4079 6983 9887
2 7229 10133 13037
$k\,=\,1\,$ j \ i 0 1 2
0 8369 11273 14177
1 11519 14423 17327
2 14669 17573 20477
$k\,=\,2\,$ j \ i 0 1 2
0 15809 18713 21617
1 18959 21863 24767
2 22109 25013 27917

## Sequences

• A113830 Leading term of a 2 X n generalized arithmetic progression (GAP) of primes with smallest last term.
• A113831 Last term of a 2 X n generalized arithmetic progression (GAP) of primes with smallest last term.
• A153412 Differences in the first known 3-by-3-by-3 generalized arithmetic progression consisting of only prime numbers.