Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
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Solving Diophantine Equation with Only Specific Variable Values Allowed
I am trying to Solve an equation like this:
$$
\sum _{i=1}^{\text{iMax}} (-i+\text{iMax}+1) n(i)=546
$$
Where all the $n(i)$ are can only take on the values 2,3,6,7, or 8.
I really have no idea how to ...
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votes
1
answer
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Finding coprime solutions to Diophantine equations
I'm hoping to use Mathematica to find solutions to Diophantine equations. Below is a toy example of something I would like to try.
Consider the case of $$x^2+y^2 = 17$$
which has the solution $x=1,y=4$...
- 225
3
votes
1
answer
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Extremum of the graph of a function has integer coordinates
Graph of the fuction $y=\dfrac{(x-26)(x+9)}{(x+14)(x+19)}$ has maximum point and minimum point are (-16,-49) and (-4,-1) whose ...
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2
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Solving a-two-variable equation in primes
How solve the following equation in Mathematica (preferably in one line) for pairs of $(x,y)$ such that $x$ and $y$ are primes?
$x^3-y^4=1$
- 115
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2
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How can I find all integer numbers so that mydistance is an integer number?
I am tring to selec two points A, B on the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 ==9^2 so that EuclideanDistance[pA,pB] is an ...
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votes
3
answers
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views
How can I select four points on a sphere to make a regular tetrahedron so that its coordinates are integer numbers?
I want to select four points lie on the sphere (x-1)^2 + (y-3)^2 + (z-5)^2 = (5* Sqrt[3])^2 so that its coordinates are integer numbers to make a regular ...
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8
votes
3
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How can I get all squares on this sphere so that its coordinates are integer numbers?
I have the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 = 15^2. I want to select all squares on this sphere so that its coordiantes are twelve different integer numbers like ...
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votes
2
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How can I find four points on a circle to make a square so that its coordinates are integer numbers?
I have a circle and I want to select four points to make a square so that its coordiantes are eight different integer numbers like this
...
- 844
1
vote
1
answer
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Parametric integer solution of $aX^2+bY^2-cZ^2=0$
Let $a,b,c\in \mathbb{Z^+}$ and unknowns $X,Y,Z\in \mathbb{Z}$. How can I get the parametric solution $(X,Y,Z)$ of equation $$aX^2+bY^2-cZ^2=0$$ in Mathematica?
In pari/gp, this solution can be ...
- 145
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2
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How can I reduce the time to compile this solve?
I want to find five integer numbers $a, b, c, d, e$ so that $a+ b+ c + d + e = 9 k$ and none of two number in which are equal. I tried.
...
- 527
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votes
4
answers
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Optimize search for rational numbers on unit circle?
Crossposted: https://community.wolfram.com/groups/-/m/t/2763509
I'd like to classify rational numbers on a unit circle by the following property: number of digits in the denominator. It relates to an ...
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Number of solutions in natural numbers $a+b+c+d=13$
I would appreciate it if somebody could help me with the following problem:
Q: To find the number of ordered pairs of natural numbers in the following equation $$a+b+c+d=13$$ we want to add the ...
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1
answer
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Solving a quadratic equations over integers efficiently
I want to solve a quadratic equation over the integers but the naive method is unbearably slow (considering how easy the problem is).
I have a positive non-degenerate quadratic form $Q$ on $\mathbb Z^...
- 135
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3
answers
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How to find integer solution of this equation?
I am trying to find pairs of integers $(x,y)$ $(x >0, y >0)$ satisfying
$$(x + y) (5 x + y)^3 + x y^3 = (5 x + y)^3 + x^2 y^3 + x y^4 $$
I tried
...
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Thue equations solving with Reduce
Could somebody tell me whether Reduce assumes (or not) the GRH when solving Thue equations?
Given its performances (related to timing) compared to PARI/GP when the "GRH assumed" flag is set ...
3
votes
1
answer
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How can I solve this system of 6 simple quadratic diophantine equations in four variables?
I generated this text file containing tuples $(\sqrt{s}, \sqrt{t}, \sqrt{u}, s,t,u,t+u,t+u-s,t-s)$ where the six variables $s,t,u,t+u,t+u-s,t-s$ are all squares. An excerpt of the data set is:
...
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Conjugate Puiseux expansions in Mathematica
In a paper "A quantitative version of Runge’s theorem on diophantine equations" ACTA ARITHMETICA
LXII.2 (1992), P. G. Walsh developed a method to solve all 2-variable diophantine equations ...
- 11
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2
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Finding all solutions to coupled algebraic equations without exhaustive enumeration
A Cambridge University mathematics interview question states:
Find all $a,b,c,d,e,f \in \mathbb{N}$ such that
$$a + b + c = d \cdot e \cdot f$$
and
$$d + e + f = a \cdot b \cdot c$$
(There are a ...
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votes
2
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Solving an equation over the integers
Consider the problem of finding all values of $n \in \mathbb{N}$ s.t. $$\sqrt{n} + \sqrt{n + 2005}$$
is an integer.
One can easily verify that $n = 1,004,004$ and $39,204$ satisfy this requirement, ...
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Find a solution to a congruence within a specified range
My problem is to find all the integer solutions of the following conguence:
$$P=0\ \text{mod}\ 4$$
where $$p=q^2∗r^2∗y^4∗z^2∗(2∗w^4+2∗r^2∗w^4∗y^4∗z^2+r^2∗w^8∗y^4∗z^2+2) + q^2+ 2∗r^6∗w^4∗y^m∗z^6∗(w+w^2+...
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Why does `FindInstance` refuse to find a second instance, in a number theory problem?
Hi I am looking for reversible Markov generating matrices of order three, whose upper (2,2) block has integer eigenvalues. There are 5 free parameters; I specified 2
...
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1
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Strange results from FrobeniusSolve
As part of pruning for my code searching for optimal addition chains I want to try and find some fast ways to discover that certain numbers are not representable in the Frobenius coin problem.
An ...
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2
answers
159
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Diophantine inequality that's not solved
Mathematica has trouble solving this Diophantine inequality:
...
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views
How do I find the solution of a system of Diophantine equations in two variables? [closed]
If the first equation is 2x+y=152, and the second is x+2y=100, and it is a fact that both variables are whole numbers and x is larger than y, what is the value of x?
- 11
2
votes
2
answers
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Solving a diophantine equation in 'large' values
Let's first discuss what I am trying to solve: I want to solve the diophantine equation stated below for relatively 'large' values of $r$.
$$\frac{a(a + 3)(a(r - 9) + (7 - r))}{12}=\frac{b (3 + b (-5 +...
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Linear diophantine equation
How to use Mathematica to solves any linear diophantine equation of the form ax+by=c, whenever it is solvable.
Such as this example, How to get the x = -165, y = 238.
Thanks!
Link: https://mathworld....
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Solving an equation in natural numbers
I am trying to solve the following equation in the Natural Numbers, with the condition $a\ge1$, $b\ge1$, and $r\ge3$:
$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$
...
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How to write this simple task of unimodular prime search in mathematica?
For testing a particular algorithm I found mathematica is the best way as it has all the tools I need. I am stuck in a number theory part and since I am not an expert in mathematica I do not know how ...
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how to solve a quadratic diophanic equation on integers and obtain various results
This quadratic di equation has more results, as it came to them.
I can only think of this
FindInstance[(4 p + 3 q - 2) (p - 1) == (6 p + 2 q) q, {p, q}, Integers]
...
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votes
2
answers
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Solving a system of equations using the data generated by PowersRepresentations and ParallelTable
First of all, it is possible to check the code that I am asking for because I know that $x=3051$ must yield at least a solution to the problem.
Well, I have the following system of equations:
Now, I ...
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Solving a system of two equations of cube integers using ParallelTable
Well, I have the following problem: I need to solve the following system of equations:
$$
\begin{cases}
n=a^3+b^3+c^3\\
\\
n=d^3+k^3+f^3
\end{cases}\tag1
$$
Where:
$n\ne a\ne b\ne c\ne d\ne k\ne f$;...
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3
votes
1
answer
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views
Find integers $a, b, c, d, m, n, p$ so equation has six distinct solutions
By hand, I found that the equation
$$ \left| -2 x+5\right| +\left| -2 x+9\right|
-x^2+7 x-16=0 $$
has six solutions $1, 2, 3, 4, 5, 6$.
Another equations
I want to find a set of integers $a, b, c, ...
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4
votes
2
answers
241
views
One more solution of the Mordell equation
Please, help to find one more solution of the Mordell equation $y^2=x^3+n,\quad n\in\mathbb{Z}$
$$y^2=x^3-307$$
Using Solve I was able to go up to ...
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9
votes
3
answers
477
views
Calculating the integral points of an elliptic curve
I asked this question on Math stachexchange. The question I have is:
Can I use Mathematica to find the (integral points on the following elliptic curve) or can I find when the number $\text{n}$ is a ...
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1
vote
1
answer
137
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Possible way to plot the solution density of diophantine equations
Well, I'm trying to investigate the density of solutions to Diophantine equations. What is a general method to describe that density?
I've two functions:
$$\varphi\left(\text{a},\text{b},\text{c}\...
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2
votes
2
answers
157
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Finding a program that can find integer solutions for large values of a variable
I've the following code:
...
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votes
0
answers
74
views
Never terminating when solving linear Diophantine systems
Say I have:
$(1):$ Set of Diophantine systems where each defines a (possibly unbounded) polyhedron.
This is solvable with existing algorithms:
$(*):$ The $(1)$ can be solved in SageMath with ...
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votes
1
answer
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views
Quadratic equation with integral coefficients [closed]
Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2+bx+c=0$ are distinct and both lie in the interval
(0,1)
(1,2)
(2,3)
(Brackets signify open interval, roots are $IN BETWEEN $ ...
4
votes
1
answer
900
views
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Why does Solve lock up when trying to solve the equation
...
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votes
4
answers
306
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FrobeniusSolve with solutions only being 0 or 1 being acceptable
I want an EFFICIENT way of solving the Frobenius Equation
a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U
where n1, n2, ..., nK are restricted to be either 0 or 1. I ...
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3
votes
4
answers
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views
How do I declare a countably infinite list of variables as being integers?
I want to do this,
Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100},
{n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ]
...
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votes
2
answers
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views
On the solvability of the Diophantine equation $a(x^2-y^2)+2bxy=1$
Take two integers $a,b$ with $\mathrm{GCD}(a,b)=1$ and $a$ odd. Consider the Diophantine equation
$$
a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z
$$
Is there any efficient way to determine whether such ...
0
votes
1
answer
137
views
Unexpected omission by Wolfram Alpha [closed]
When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2
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How can I reduce the time to run this code?
I am trying to find the integers $ a, b, c, d \in [-15, -1] \cup [1, 15] $ so that the equation $ \left| x^2 + a x + b \right| = c x + d $ has four distinct integeral solutions different from $ 0 $.
...
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5
votes
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answer
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views
Pythagorean quadruples
I am very interested in the problem that deals with the Pythagorean quadrruples, which are listed in rosettacode. Unfortunately, I have not been able to find any way to compute them with Mathematica. ...
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3
votes
1
answer
108
views
Guess Diophantine equation from its solutions
Take for example the equation
$$
n^2=x^2+y^2+1
$$
where $x,y$ are integers.
This equation admits solutions only for some values of $n$, to wit,
$$
n=1,3,9,17,19,33,35,51,73,81,99,\dots
$$
Can we ...
1
vote
1
answer
430
views
How do we show reduction steps?
This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper.
...
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4
votes
3
answers
644
views
Finding Smallest Positive Solution to Diophantine Equation
Mathematica can solve $v^2- d u^2=4$ quickly, even for nonsmall $d$:
d = 400004;
Reduce[v^2 - d u^2 == 4, Integers]
d = 400012;
Reduce[v^2 - d u^2 == 4, Integers]
...
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3
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Solving a system of diophantine equations from a mathematical competition
There is a problem from the $66th$ Putnam Mathematical Competition, $B2$
Find all positive integers $n,k_1,k_2,...,k_n$ such that $$k_1+\cdots+k_n=5n-4$$
and
$$\frac{1}{k_1}+\cdots+\frac{1}{...
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2
answers
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Solving a diophantine equation with three variables
Solve this equation using Mathematica:
$2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.