Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
75
questions
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votes
1
answer
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views
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
4
votes
3
answers
441
views
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
...
- 1,851
1
vote
0
answers
66
views
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
136
views
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:
...
- 315
1
vote
0
answers
81
views
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
0
votes
2
answers
136
views
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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5
votes
2
answers
468
views
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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0
votes
1
answer
41
views
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 $$p=q^2∗r^2∗y^4∗z^2∗(2∗w^4+2∗r^2∗w^4∗y^4∗z^2+r^2∗w^8∗y^4∗z^2+2) + q^2+ 2∗r^6∗w^4∗y^m∗z^6∗(w+w^2+...
- 121
1
vote
0
answers
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Why does `FindInstance` refuse to find a second instance, in a number theory problem?
Hi I am looking for reversible Markov generating matrices of order three, whose upper (2,2) block has integer eigenvalues. There are 5 free parameters; I specified 2
...
- 1,202
2
votes
1
answer
65
views
Strange results from FrobeniusSolve
As part of pruning for my code searching for optimal addition chains I want to try and find some fast ways to discover that certain numbers are not representable in the Frobenius coin problem.
An ...
- 123
3
votes
2
answers
123
views
Diophantine inequality that's not solved
Mathematica has trouble solving this Diophantine inequality:
...
- 416
0
votes
1
answer
48
views
How do I find the solution of a system of Diophantine equations in two variables? [closed]
If the first equation is 2x+y=152, and the second is x+2y=100, and it is a fact that both variables are whole numbers and x is larger than y, what is the value of x?
- 11
2
votes
2
answers
106
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Solving a diophantine equation in 'large' values
Let's first discuss what I am trying to solve: I want to solve the diophantine equation stated below for relatively 'large' values of $r$.
$$\frac{a(a + 3)(a(r - 9) + (7 - r))}{12}=\frac{b (3 + b (-5 +...
- 1,765
0
votes
1
answer
632
views
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....
- 47
3
votes
4
answers
489
views
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 $r\ge3$:
$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$
...
- 1,765
2
votes
1
answer
85
views
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 ...
- 135
0
votes
1
answer
49
views
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]
...
- 2,175
0
votes
2
answers
107
views
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 ...
- 1,765
1
vote
1
answer
78
views
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:
$n\ne a\ne b\ne c\ne d\ne k\ne f$;...
- 1,765
3
votes
1
answer
270
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, ...
- 1,851
4
votes
2
answers
212
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 ...
- 16.3k
8
votes
3
answers
387
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 ...
- 1,765
1
vote
1
answer
118
views
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:
$$\varphi\left(\text{a},\text{b},\text{c}\...
- 1,765
2
votes
2
answers
153
views
Finding a program that can find integer solutions for large values of a variable
I've the following code:
...
- 1,765
2
votes
0
answers
67
views
Never terminating when solving linear Diophantine systems
Say I have:
$(1):$ Set of Diophantine systems where each defines a (possibly unbounded) polyhedron.
This is solvable with existing algorithms:
$(*):$ The $(1)$ can be solved in SageMath with ...
- 622
-1
votes
1
answer
70
views
Quadratic equation with integral coefficients [closed]
Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2+bx+c=0$ are distinct and both lie in the interval
(0,1)
(1,2)
(2,3)
(Brackets signify open interval, roots are $IN BETWEEN $ ...
4
votes
1
answer
895
views
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Why does Solve lock up when trying to solve the equation
...
- 43
2
votes
4
answers
276
views
FrobeniusSolve with solutions only being 0 or 1 being acceptable
I want an EFFICIENT way of solving the Frobenius Equation
a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U
where n1, n2, ..., nK are restricted to be either 0 or 1. I ...
- 1,430
3
votes
4
answers
218
views
How do I declare a countably infinite list of variables as being integers?
I want to do this,
Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100},
{n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ]
...
- 1,430
2
votes
2
answers
135
views
On the solvability of the Diophantine equation $a(x^2-y^2)+2bxy=1$
Take two integers $a,b$ with $\mathrm{GCD}(a,b)=1$ and $a$ odd. Consider the Diophantine equation
$$
a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z
$$
Is there any efficient way to determine whether such ...
0
votes
1
answer
132
views
Unexpected omission by Wolfram Alpha [closed]
When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2
- 111
4
votes
2
answers
769
views
How can I reduce the time to run this code?
I am trying to find the integers $ a, b, c, d \in [-15, -1] \cup [1, 15] $ so that the equation $ \left| x^2 + a x + b \right| = c x + d $ has four distinct integeral solutions different from $ 0 $.
...
- 1,851
3
votes
1
answer
105
views
Guess Diophantine equation from its solutions
Take for example the equation
$$
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
$$
Can we ...
1
vote
1
answer
315
views
How do we show reduction steps?
This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper.
...
- 2,298
4
votes
3
answers
566
views
Finding Smallest Positive Solution to Diophantine Equation
Mathematica can solve $v^2- d u^2=4$ quickly, even for nonsmall $d$:
d = 400004;
Reduce[v^2 - d u^2 == 4, Integers]
d = 400012;
Reduce[v^2 - d u^2 == 4, Integers]
...
- 508
4
votes
3
answers
714
views
Solving a system of diophantine equations from a mathematical competition
There is a problem from the $66th$ Putnam Mathematical Competition, $B2$
Find all positive integers $n,k_1,k_2,...,k_n$ such that $$k_1+\cdots+k_n=5n-4$$
and
$$\frac{1}{k_1}+\cdots+\frac{1}{...
- 8,668
-3
votes
2
answers
417
views
Solving a diophantine equation with three variables
Solve this equation using Mathematica:
$2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.
7
votes
2
answers
208
views
Unexpected behaviour with FindInstance
Evaluating
FindInstance[a/(b + c) + b/(a + c) + c/(a + b) == 4, {a, b, c}, Integers]
does what I'd expect it to do: find a solution to the equation in integers.
...
- 412
2
votes
1
answer
224
views
Storing Multiple solutions from Solve as rows of a matrix
I am trying to find all the solutions to a Diophantine equation (only integer valued solutions) using Solve[], however there are multiple and even many solutions sometimes.
How can I store these ...
- 181
1
vote
2
answers
555
views
determine the coefficients of a quadratic equation with "Solve"
Assume we want to determine the coefficients (integer) of a quadratic equation by having some information about the input and output values. If I enter the input and output individually before the ...
- 808
5
votes
1
answer
619
views
Solving a Diophantine equation with a large solution
I am trying to solve the following Diophantine equation with Mathematica:
$\frac{x}{y+z}+\frac{z}{x+y}+\frac{y}{x+z}=4$
It is known that there are three positive numbers that satisfy the equation ...
- 181
2
votes
0
answers
78
views
Introducing constraints in FrobeniusSolve
I have a knapsack problem. Say I have N possible items $x_i$. I would like to know:
$ \sum_{i=0}^{2} c_i x_i = W$
With the following constraint:
$c_i=3 \lor 4$
As an example suppose the set $x_i=...
- 2,078
2
votes
0
answers
209
views
How to test solvability of a Diophantine equation?
When applying the BRC theorem I need to test if an equation has a non-zero solution in integers.
The equation is of form:
$$x^2 + by^2 + cz^2 = 0$$
We could try to directly check if this equation ...
- 325
2
votes
2
answers
325
views
Integer partitions without repetitions
What combination of numbers makes a specific sum?
The code below is not very effective, because it also gives answers in which a number is used more than once even though it was given in the list ...
- 321
5
votes
2
answers
149
views
Finding all solutions in the Roth's theorem
Roth's theorem. For all algebraic irrational $\alpha$
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$
with $\epsilon>0$, has finitely many ...
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16
votes
5
answers
2k
views
Puzzle — 20 People to consume 20 units of food under constraints
I am learning Mathematica because I love it. I also love solving puzzles so I think it would be a nice way to learn Mathematica through puzzles. This is first puzzle in series I intend to solve.
So ...
- 321
0
votes
1
answer
709
views
Solving Diophantine equations
Given a complex number $z$ and a positive integer $n$, I would like to be able to find integer solutions $\alpha,\beta,\gamma$ to the Diophantine equation
$$0 < a^2\vert z \vert^2 + \beta\textrm{ ...
- 119
4
votes
3
answers
391
views
find and count the number of solutions without multiplicity in Solve?
I would like to solve a Diophantine equation and find its solution, but I need only count one time for each $a$, i.e., when for some $a$ it found some $x,y,z$, then go to the next $a$. more precisely ...
- 808
2
votes
2
answers
338
views
Solve an exponential equation in integers
Find two integers x and y such that $x^y +y^x = 94032$.
I have used
...
- 2,175
21
votes
3
answers
991
views
Partitioning a number into consecutive integers
Consider n=45; then
$$1+2+3+4+5+6+7+8+9=45$$
$$5+6+7+8+9+10=45$$
$$7+8+9+10+11=45$$
$$14+15+16=45$$
$$22+23=45$$
Question: how to find all representations of a ...
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