Dr. Sameen Ahmed Khan
([email protected])
(Visiting Card,
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Assistant Professor,
Department of Mathematics and Sciences
College of Arts and Applied Sciences (CAAS)
Dhofar University
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Post Box No. 2509,
Postal Code
211
Salalah
Dhofar
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Oman
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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.
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}.
List of 37+ Writeups from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo). | List of 21+ Writeups from the LANL E-Print archive (see the Atom Feeds). |
Some Publications from the American Mathematical Society (AMS, Logo) MathSciNet (Logo). | List of 260+ Writeups from the Google Scholar (Logo). |
Research in Charged-Particle Beam Optics.
Research in Light Beam Optics.
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Quadricmeter is the instrument devised to identify (distinguish) and measure the various
parameters (axis, foci, latera recta, directrix, etc.,) completely characterizing the important
class of surfaces known as the quadratic surfaces. Quadratic surfaces (also known as quadrics)
include a wide range of commonly encountered surfaces including, cone, cylinder, ellipsoid,
elliptic cone, elliptic cylinder, elliptic hyperboloid, elliptic paraboloid, hyperbolic cylinder,
hyperbolic paraboloid, paraboloid, sphere, and spheroid. Quadricmeter is a generalized form of
the conventional spherometer and the lesser known cylindrometer (also known as the "Cylindro-Spherometer"
and "Sphero-Cylindrometer").
With a conventional spherometer it was possible only to measure the radii of spherical surfaces.
Cylindrometer can measure the radii of curvature of a cylindrical surface in addition to the spherical
surface. In both the spherometer and the cylindrometer one assumes the surface to be either spherical
or cylindrical respectively. In the case of the quadricmeter, there are no such assumptions.
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