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?
> 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
Konstantin Smirnov wrote: > 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.
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:
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".
> > 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
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:
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
>>> 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)
> 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. And beeing able to read, write and calculate is a luxus nowadays while we have such able politicians ... and entertaining folks in newsgroups ... -- RobertZ
> 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?
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
On Oct 6, 4:19 pm, Konstantin Smirnov <konstantin.e.smir...@gmail.com> wrote:
> 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?
> In article <4aeb0c7f$0$6590$9b4e6...@newsspool3.arcor-online.net>, > Robert Z <t9353...@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?
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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