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| Konstantin Smirnov... |
 Posted: Tue Oct 06, 2009 11:19 am |
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Guest
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Dear number theorists,
what you can say about modern applications of Diophantine equations,
especially having large numbers solutions (>10^10-10^20)? In general,
number
theory is a standalone fundamental area, and Diophantine equations can
be
investigated only as a part of work in number theory. But can you
suggest
any significant applications that use Diophantine equations? If you
work
with such applications, please post what type of equations you use in
this area.
Mostly I am interested in equations like x^k+y^l+z^m=t^n. Is there any
practical benefit from them or such equations are only of theoretical
interest? What are main areas besides cryptography and coding?
Also what is the fastest program for search of solutions of such
equations? Mathematica, Maple, Pari GP or smth else?
Thanks
Konstantin |
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| Mensanator... |
| Posted: Tue Oct 06, 2009 12:40 pm |
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On Oct 6, 4:40 pm, "maryK" <inva... at (no spam) invilidinvalid.com> wrote:
[quote:26152cada0]"Konstantin Smirnov" <konstantin.e.smir... at (no spam) gmail.com> wrote in message
news:8cbfba3b-d0fb-4dc9-a3f4-ea0cafcf3634 at (no spam) a6g2000vbp.googlegroups.com...
Dear number theorists,
what you can say about modern applications of Diophantine equations,
especially having large numbers solutions (>10^10-10^20)? In general,
number
theory is a standalone fundamental area, and Diophantine equations can
be
investigated only as a part of work in number theory. But can you
suggest
any significant applications that use Diophantine equations? If you
work
with such applications, please post what type of equations you use in
this area.
Mostly I am interested in equations like x^k+y^l+z^m=t^n. Is there any
practical benefit from them or such equations are only of theoretical
interest? What are main areas besides cryptography and coding?
Also what is the fastest program for search of solutions of such
equations? Mathematica, Maple, Pari GP or smth else?
Thanks
Konstantin
not much use for them
mathematical curiosities mostly
[/quote:26152cada0]
Linear ones are useful, a form of which is the Hailstone Function
X*a - Z
g = -------
Y
in the Collatz Conjecture (where X,Y,Z are constants and we want
integer solutions for g and a).
One nice thing is that every Yth 1st generation solution is a second
generation solution, starting from the a1_kth solution, every third
generation solution is the Yth second generation solution starting
from
the a2_mth solution, every fourth generation solution is the a3_nth
solution, etc.
If you're lucky, and k = m = n = ..., then a closed form equation can
be derived such as this one for the ith, kth Generation Type [1,2]
Mersenne Hailstone:
Type12MH = 2**(6*((i-1)*9**(k-1)+(9**(k-1)-1)//2+1)-1)-1
This was derived from solving the linear congruence X*a == Z (mod Y)
which can be used to find solutions the the linear Diophantine
equation
given above. BTW, Type12MH(6,1) has 53338 decimal digits.
If you are unlucky, k != m != n ...
But with a little cleverness, you can make a recursive function with
generation one being X*a == Z (mod Y) which can be solved by
a = gmpy.invert(X,Y) * Z % Y
and to get higher generations, merely use
a = (((gmpy.invert(xyz[1]-xyz[0],xyz[1]**(k-1))*(xyz[1]**(k-1)-
prev_gen[2]))_
% xyz[1]**(k-1))//xyz[1]**(k-2))*xyz[1]**(k-1) + prev_gen[3]
which solves the multigenerational linear Diophantine equation where
k,m,n,etc. are different.
For example
[quote:26152cada0]sv = [i for i in range(666)] # sequence of 666 consecutive numbers
xyz = cf.calc_xyz(sv)
a = cf.geni(666,666,xyz) # find the 666th instance of the 666th generation
gmpy.numdigits(a)
211634 # it has that many digits![/quote:26152cada0] |
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| maryK... |
| Posted: Tue Oct 06, 2009 3:40 pm |
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Guest
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"Konstantin Smirnov" <konstantin.e.smirnov at (no spam) gmail.com> wrote in message
news:8cbfba3b-d0fb-4dc9-a3f4-ea0cafcf3634 at (no spam) a6g2000vbp.googlegroups.com...
[quote:1025fd4cdb]Dear number theorists,
what you can say about modern applications of Diophantine equations,
especially having large numbers solutions (>10^10-10^20)? In general,
number
theory is a standalone fundamental area, and Diophantine equations can
be
investigated only as a part of work in number theory. But can you
suggest
any significant applications that use Diophantine equations? If you
work
with such applications, please post what type of equations you use in
this area.
Mostly I am interested in equations like x^k+y^l+z^m=t^n. Is there any
practical benefit from them or such equations are only of theoretical
interest? What are main areas besides cryptography and coding?
Also what is the fastest program for search of solutions of such
equations? Mathematica, Maple, Pari GP or smth else?
Thanks
Konstantin
[/quote:1025fd4cdb]
not much use for them
mathematical curiosities mostly |
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| I.N. Galidakis... |
| Posted: Tue Oct 06, 2009 3:59 pm |
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Konstantin Smirnov wrote:
[quote:335f468472]Dear number theorists,
what you can say about modern applications of Diophantine equations,
especially having large numbers solutions (>10^10-10^20)? In general,
number
theory is a standalone fundamental area, and Diophantine equations can
be
investigated only as a part of work in number theory. But can you
suggest
any significant applications that use Diophantine equations? If you
work
with such applications, please post what type of equations you use in
this area.
[/quote:335f468472]
Simple Diophantine equations have an application in video and lighting
technology. One area which I am aware of, is intereference of flickering light
sources and video capturing devices, or more generally, interference between
devices which capture scenery at a fixed rate of frames per second, and lights,
when the scenery is illuminated by lights whose frequency depends on the mains
frequency non-trivially:
http://ioannis.virtualcomposer2000.com/math/video.html
This is one of the solutions to the particular interference problem. The other
involves using ballasts which use lagging techniques to turn flickering lights
on and off in groups of n.
More is available if you Google "stroboscopic effect".
[quote:335f468472]Thanks
Konstantin
--[/quote:335f468472]
Ioannis |
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| Robert Z... |
| Posted: Fri Oct 30, 2009 9:56 am |
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Konstantin Smirnov schrieb:
[quote]Dear number theorists,
what you can say about modern applications of Diophantine equations,
especially having large numbers solutions (>10^10-10^20)? In general,
number
theory is a standalone fundamental area, and Diophantine equations can
be
investigated only as a part of work in number theory. But can you
suggest
any significant applications that use Diophantine equations? If you
work
with such applications, please post what type of equations you use in
this area.
Mostly I am interested in equations like x^k+y^l+z^m=t^n. Is there any
practical benefit from them or such equations are only of theoretical
interest?
[/quote]
Substituting
x:=a
k:=2
y:=b
l:=2
z:=0
t:=c
n:=2
Sure this equation is only of "theoretical interest". No one needs a
right angle. And beeing able to read, write and calculate is a luxus
nowadays while we have such able politicians ... and entertaining folks
in newsgroups ...
--
RobertZ |
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| Gerry Myerson... |
| Posted: Mon Nov 02, 2009 12:26 am |
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In article <4aeb0c7f$0$6590$9b4e6d93 at (no spam) newsspool3.arcor-online.net>,
Robert Z <t9353396 at (no spam) arcor.de> wrote:
[quote]Konstantin Smirnov schrieb:
Dear number theorists,
what you can say about modern applications of Diophantine
equations, especially having large numbers solutions
(>10^10-10^20)? In general, number theory is a standalone
fundamental area, and Diophantine equations can be investigated
only as a part of work in number theory. But can you suggest any
significant applications that use Diophantine equations? If you
work with such applications, please post what type of equations you
use in this area. Mostly I am interested in equations like
x^k+y^l+z^m=t^n. Is there any practical benefit from them or such
equations are only of theoretical interest?
Substituting
x:=a
k:=2
y:=b
l:=2
z:=0
t:=c
n:=2
Sure this equation is only of "theoretical interest". No one needs a
right angle.
[/quote]
OK, we need right angles. Do we need right-angle triangles
in which each side is an integer?
--
Gerry Myerson (gerry at (no spam) maths.mq.edi.ai) (i -> u for email) |
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| Pubkeybreaker... |
| Posted: Mon Nov 02, 2009 3:19 am |
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Guest
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On Oct 6, 4:19 pm, Konstantin Smirnov <konstantin.e.smir... at (no spam) gmail.com>
wrote:
[quote]Dear number theorists,
what you can say about modern applications of Diophantine equations,
especially having large numbers solutions (>10^10-10^20)? In general,
number
theory is a standalone fundamental area, and Diophantine equations can
be
investigated only as a part of work in number theory. But can you
suggest
any significant applications that use Diophantine equations?
[/quote]
Yes. Cryptography. As you stated below. |
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| Robert Z... |
| Posted: Tue Nov 03, 2009 3:10 pm |
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Gerry Myerson schrieb:
[quote]In article <4aeb0c7f$0$6590$9b4e6d93 at (no spam) newsspool3.arcor-online.net>,
Robert Z <t9353396 at (no spam) arcor.de> wrote:
Konstantin Smirnov schrieb:
Dear number theorists,
what you can say about modern applications of Diophantine
equations, especially having large numbers solutions
(>10^10-10^20)? In general, number theory is a standalone
fundamental area, and Diophantine equations can be investigated
only as a part of work in number theory. But can you suggest any
significant applications that use Diophantine equations? If you
work with such applications, please post what type of equations you
use in this area. Mostly I am interested in equations like
x^k+y^l+z^m=t^n. Is there any practical benefit from them or such
equations are only of theoretical interest?
Substituting
x:=a
k:=2
y:=b
l:=2
z:=0
t:=c
n:=2
Sure this equation is only of "theoretical interest". No one needs a
right angle.
OK, we need right angles. Do we need right-angle triangles
in which each side is an integer?
[/quote]
Thats a philosophic question.
Wikipedia says: "In mathematics, a Diophantine equation is an
indeterminate polynomial equation that allows the variables to be
integers only."
But the way from a non-integer right-angle triangle to an integer
right-angle triangle would be as interesting as the way from real
numbers to integers in the equation p*q=n.
--
RobertZ |
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