Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
70
questions
1
vote
0answers
72 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 ...
0
votes
2answers
119 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 ...
0
votes
1answer
29 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 $$p=q^2∗r^2∗y^4∗z^2∗(2∗w^4+2∗r^2∗w^4∗y^4∗z^2+r^2∗w^8∗y^4∗z^2+2) + q^2+ 2∗r^6∗w^4∗y^m∗z^6∗(w+w^2+...
1
vote
0answers
48 views
Why does `FindInstance` refuse to find a second instance, in a number theory problem?
Hi I am looking for reversible Markov generating matrices of order three, whose upper (2,2) block has integer eigenvalues. There are 5 free parameters; I specified 2
...
2
votes
1answer
60 views
Strange results from FrobeniusSolve
As part of pruning for my code searching for optimal addition chains I want to try and find some fast ways to discover that certain numbers are not representable in the Frobenius coin problem.
An ...
3
votes
2answers
109 views
Diophantine inequality that's not solved
Mathematica has trouble solving this Diophantine inequality:
...
0
votes
1answer
44 views
How do I find the solution of a system of Diophantine equations in two variables? [closed]
If the first equation is 2x+y=152, and the second is x+2y=100, and it is a fact that both variables are whole numbers and x is larger than y, what is the value of x?
2
votes
2answers
100 views
Solving a diophantine equation in 'large' values
Let's first discuss what I am trying to solve: I want to solve the diophantine equation stated below for relatively 'large' values of $r$.
$$\frac{a(a + 3)(a(r - 9) + (7 - r))}{12}=\frac{b (3 + b (-5 +...
0
votes
1answer
195 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....
3
votes
4answers
396 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 $r\ge3$:
$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$
...
2
votes
1answer
82 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 ...
0
votes
1answer
43 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]
...
0
votes
2answers
103 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
vote
1answer
75 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:
$n\ne a\ne b\ne c\ne d\ne k\ne f$;...
3
votes
1answer
249 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, ...
4
votes
2answers
198 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 ...
7
votes
3answers
324 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
vote
1answer
104 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:
$$\varphi\left(\text{a},\text{b},\text{c}\...
2
votes
2answers
142 views
Finding a program that can find integer solutions for large values of a variable
I've the following code:
...
2
votes
0answers
62 views
Never terminating when solving linear Diophantine systems
Say I have:
$(1):$ Set of Diophantine systems where each defines a (possibly unbounded) polyhedron.
This is solvable with existing algorithms:
$(*):$ The $(1)$ can be solved in SageMath with ...
-1
votes
1answer
66 views
Quadratic equation with integral coefficients [closed]
Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2+bx+c=0$ are distinct and both lie in the interval
(0,1)
(1,2)
(2,3)
(Brackets signify open interval, roots are $IN BETWEEN $ ...
4
votes
1answer
887 views
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Why does Solve lock up when trying to solve the equation
...
2
votes
4answers
254 views
FrobeniusSolve with solutions only being 0 or 1 being acceptable
I want an EFFICIENT way of solving the Frobenius Equation
a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U
where n1, n2, ..., nK are restricted to be either 0 or 1. I ...
3
votes
4answers
195 views
How do I declare a countably infinite list of variables as being integers?
I want to do this,
Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100},
{n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ]
...
2
votes
2answers
131 views
On the solvability of the Diophantine equation $a(x^2-y^2)+2bxy=1$
Take two integers $a,b$ with $\mathrm{GCD}(a,b)=1$ and $a$ odd. Consider the Diophantine equation
$$
a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z
$$
Is there any efficient way to determine whether such ...
asked Nov 14 '18 at 17:45
AccidentalFourierTransform
9,1871 gold badge16 silver badges48 bronze badges
0
votes
1answer
131 views
Unexpected omission by Wolfram Alpha [closed]
When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2
4
votes
2answers
764 views
How can I reduce the time to run this code?
I am trying to find the integers $ a, b, c, d \in [-15, -1] \cup [1, 15] $ so that the equation $ \left| x^2 + a x + b \right| = c x + d $ has four distinct integeral solutions different from $ 0 $.
...
3
votes
1answer
102 views
Guess Diophantine equation from its solutions
Take for example the equation
$$
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
$$
Can we ...
asked Aug 2 '18 at 21:07
AccidentalFourierTransform
9,1871 gold badge16 silver badges48 bronze badges
1
vote
1answer
243 views
How do we show reduction steps?
This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper.
...
4
votes
3answers
477 views
Finding Smallest Positive Solution to Diophantine Equation
Mathematica can solve $v^2- d u^2=4$ quickly, even for nonsmall $d$:
d = 400004;
Reduce[v^2 - d u^2 == 4, Integers]
d = 400012;
Reduce[v^2 - d u^2 == 4, Integers]
...
3
votes
3answers
672 views
Solving a system of diophantine equations from a mathematical competition
There is a problem from the $66th$ Putnam Mathematical Competition, $B2$
Find all positive integers $n,k_1,k_2,...,k_n$ such that $$k_1+\cdots+k_n=5n-4$$
and
$$\frac{1}{k_1}+\cdots+\frac{1}{...
-3
votes
2answers
390 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
2answers
192 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.
...
2
votes
1answer
175 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 ...
1
vote
2answers
538 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 ...
5
votes
1answer
578 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 ...
2
votes
0answers
72 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
votes
0answers
182 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 ...
2
votes
2answers
293 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 ...
5
votes
2answers
143 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 ...
16
votes
5answers
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 ...
0
votes
1answer
600 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
$$0 < a^2\vert z \vert^2 + \beta\textrm{ ...
4
votes
3answers
317 views
find and count the number of solutions without multiplicity in Solve?
I would like to solve a Diophantine equation and find its solution, but I need only count one time for each $a$, i.e., when for some $a$ it found some $x,y,z$, then go to the next $a$. more precisely ...
2
votes
2answers
322 views
Solve an exponential equation in integers
Find two integers x and y such that $x^y +y^x = 94032$.
I have used
...
20
votes
3answers
948 views
Partitioning a number into consecutive integers
Consider n=45; then
$$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$$
Question: how to find all representations of a ...
1
vote
0answers
97 views
diophantine linear equation with condition on the sum of coefficients [closed]
I am trying to solve the following diophantine equation
x*a + y*b == c
where a,b,c are integers and the absolute value of their sums is, for instance, 4. Then I ...
2
votes
2answers
365 views
how to find sum of variable c1+c2+c3 for expression combinations
how to find sum of variable c1 + c2 + c3 for expression combinations:
...
5
votes
3answers
216 views
On Solutions of Diophantine Equation
By Mathematica, I try to obtain solutions of Diophantine equations such as:
$$F_{n_1}F_{n_2}\cdots F_{n_k}+1=F_t$$
where the sequence $\{F_n\}$ is the Fibonacci sequence,$n_1 < n_2 < \cdots &...
6
votes
1answer
128 views
Find out whether solutions exist rather than full-blown solution search
I have some quadratics, and I am trying to find out whether there exist solutions in the integers. The following tells me that the first does, wheraeas the second doesn't:
...
4
votes
4answers
1k views
How to merge permutations obtained from Solve on multiple variables?
When I compute the following expression to find integer solutions of the equation (x^2 + y^2 + z^2 == 14^2)
...
