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A geometric-arithmetic progression of primes is a set of k primes
(denoted by GAP-k) of the form p_{1}*r ^{j} + j*d
for fixed p_{1}, r and d and consecutive j,
from j = 0 to k - 1.
i.e, {p_{1}, p_{1}*r + d, p_{1}*r ^{2} + 2 d,
p_{1}* r ^{3} + 3 d, ...}.
For example 3, 17, 79 is a 3-term geometric-arithmetic progression
(i.e, a GAP-3) with a = p_{1} = 3, r = 5 and d = 2.
A GAP-k is said to be minimal if the minimal start p_{1} and
the minimal ratio r are equal, i.e, p_{1} = r = p, where p
is the smallest prime ≥ k.
Such GAPs have the form p*p ^{j} + j*d.
Minimal GAPs with different differences, d do exist. For example, the minimal GAP-5
(p_{1} = r = 5) has the
possible differences, 84, 114, 138, 168, ... (see the Sequence A209204)
and the minimal
GAP-6 (p_{1} = r = 7) has the possible differences,
144, 1494, 1740, 2040, .... (see the Sequence A209205).
The following article gives the conditions under which, a GAP-k is a
set of k primes in geometric-arithmetic progression.
Sameen Ahmed Khan,
Primes in Geometric-Arithmetic Progression,
19 pages,
LANL
E-Print Archive:
http://arxiv.org/abs/1203.2083/.
Bibliographic Code:
2012arXiv1203.2083K
(Friday the 09 March 2012).
The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
The exceptional cases (for k < = 7) are k = 2, k = 3, k = 5 and k = 7.
For k = 2, we have d = 1 and there is ONLY one AP-2 with this difference: {2, 3}.
For k = 3, we have d = 2 and there is ONLY one AP-3 with this difference: {3, 5, 7}.
For k = 4, we have d = 4# = 6 and AP-4 is {5, 11, 17, 23} and is not unique.
The first primes is the Sequence A023271:
5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, ...
For k = 5, we have d = 3# = 6 and there is ONLY one AP-5 with this difference: {5, 11, 17, 23, 29}.
For k = 6, we have d = 6# = 30 and AP-6 is {7, 37, 67, 97, 127, 157} and is not unique.
The first primes is the Sequence A156204:
7, 107, 359, 541, 2221, 6673, 7457, 10103, 25643, 26861, 27337, 35051, 56149, ...
For k = 7, we have d = 5*5# = 150 and there is ONLY one AP-7 with this difference:
{7, 157, 307, 457, 607, 757, 907}.
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