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# Generalized arithmetic progressions

A **generalized arithmetic progression** (**GAP**) (**multiple arithmetic progression**, -**dimensional arithmetic progression**) is defined as

where the are fixed.

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

## Contents

## Generalized arithmetic progressions (GAP) of primes

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

Here is the beginning of Granville's table:^{[1]} (Cf. A113830, A113831)

**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:

**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

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)

**j \ i****0****1****2****0***929**3833**6737***1***4079**6983**9887***2***7229**10133**13037*

**j \ i****0****1****2****0***8369**11273**14177***1***11519**14423**17327***2***14669**17573**20477*

**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.

## See also

## Notes

- ↑ Andrew Granville, Prime number patterns.