Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
60
questions
0
votes
1answer
40 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]
...
1
vote
0answers
60 views
7X7 matrix of distinct cubed integers with identical row and column sums
Problem: find a 7X7 matrix of distinct cubed integers with identical row and column sums.
Well, I have the following code (that came from @kglr from my previous question):
...
0
votes
2answers
88 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
70 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
205 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
187 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
236 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 ...
0
votes
1answer
80 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:
$$\varphi\left(\text{a},\text{b},\text{c}\...
2
votes
2answers
131 views
Finding a program that can find integer solutions for large values of a variable
I've the following code:
...
2
votes
0answers
59 views
Never terminating when solving linear Diophantine systems
Say I have:
$(1):$ Set of Diophantine systems where each defines a (possibly unbounded) polyhedron.
This is solvable with existing algorithms:
$(*):$ The $(1)$ can be solved in SageMath with ...
0
votes
0answers
58 views
LinearProgramming with constraint on result
I need to solve a bunch of Diophantine Equations that takes so looong with FindInstance/Reduce, so I tried LinearProgramming and it's very fast. Here is a simple one:
...
-1
votes
1answer
56 views
Quadratic equation with integral coefficients [closed]
Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2+bx+c=0$ are distinct and both lie in the interval
(0,1)
(1,2)
(2,3)
(Brackets signify open interval, roots are $IN BETWEEN $ ...
4
votes
1answer
879 views
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Why does Solve lock up when trying to solve the equation
...
2
votes
4answers
236 views
FrobeniusSolve with solutions only being 0 or 1 being acceptable
I want an EFFICIENT way of solving the Frobenius Equation
a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U
where n1, n2, ..., nK are restricted to be either 0 or 1. I ...
3
votes
4answers
181 views
How do I declare a countably infinite list of variables as being integers?
I want to do this,
Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100},
{n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ]
...
2
votes
2answers
125 views
On the solvability of the Diophantine equation $a(x^2-y^2)+2bxy=1$
Take two integers $a,b$ with $\mathrm{GCD}(a,b)=1$ and $a$ odd. Consider the Diophantine equation
$$
a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z
$$
Is there any efficient way to determine whether such ...
asked Nov 14 '18 at 17:45
AccidentalFourierTransform
8,8641 gold badge14 silver badges48 bronze badges
0
votes
1answer
123 views
Unexpected omission by Wolfram Alpha [closed]
When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2
4
votes
2answers
752 views
How can I reduce the time to run this code?
I am trying to find the integers $ a, b, c, d \in [-15, -1] \cup [1, 15] $ so that the equation $ \left| x^2 + a x + b \right| = c x + d $ has four distinct integeral solutions different from $ 0 $.
...
3
votes
1answer
101 views
Guess Diophantine equation from its solutions
Take for example the equation
$$
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
$$
Can we ...
asked Aug 2 '18 at 21:07
AccidentalFourierTransform
8,8641 gold badge14 silver badges48 bronze badges
1
vote
1answer
145 views
How do we show reduction steps?
This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper.
...
4
votes
3answers
386 views
Finding Smallest Positive Solution to Diophantine Equation
Mathematica can solve $v^2- d u^2=4$ quickly, even for nonsmall $d$:
d = 400004;
Reduce[v^2 - d u^2 == 4, Integers]
d = 400012;
Reduce[v^2 - d u^2 == 4, Integers]
...
3
votes
3answers
660 views
Solving a system of diophantine equations from a mathematical competition
There is a problem from the $66th$ Putnam Mathematical Competition, $B2$
Find all positive integers $n,k_1,k_2,...,k_n$ such that $$k_1+\cdots+k_n=5n-4$$
and
$$\frac{1}{k_1}+\cdots+\frac{1}{...
-3
votes
2answers
352 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
178 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
154 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 ...
5
votes
1answer
496 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
68 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
153 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
267 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
131 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
447 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
$$0 < a^2\vert z \vert^2 + \beta\textrm{ ...
4
votes
3answers
245 views
find and count the number of solutions without multiplicity in Solve?
I would like to solve a Diophantine equation and find its solution, but I need only count one time for each $a$, i.e., when for some $a$ it found some $x,y,z$, then go to the next $a$. more precisely ...
2
votes
2answers
275 views
Solve an exponential equation in integers
Find two integers x and y such that $x^y +y^x = 94032$.
I have used
...
20
votes
3answers
892 views
Partitioning a number into consecutive integers
Consider n=45; then
$$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$$
Question: how to find all representations of a ...
1
vote
0answers
92 views
diophantine linear equation with condition on the sum of coefficients [closed]
I am trying to solve the following diophantine equation
x*a + y*b == c
where a,b,c are integers and the absolute value of their sums is, for instance, 4. Then I ...
2
votes
2answers
330 views
how to find sum of variable c1+c2+c3 for expression combinations
how to find sum of variable c1 + c2 + c3 for expression combinations:
...
5
votes
3answers
198 views
On Solutions of Diophantine Equation
By Mathematica, I try to obtain solutions of Diophantine equations such as:
$$F_{n_1}F_{n_2}\cdots F_{n_k}+1=F_t$$
where the sequence $\{F_n\}$ is the Fibonacci sequence,$n_1 < n_2 < \cdots &...
6
votes
1answer
118 views
Find out whether solutions exist rather than full-blown solution search
I have some quadratics, and I am trying to find out whether there exist solutions in the integers. The following tells me that the first does, wheraeas the second doesn't:
...
3
votes
4answers
1k views
How to merge permutations obtained from Solve on multiple variables?
When I compute the following expression to find integer solutions of the equation (x^2 + y^2 + z^2 == 14^2)
...
3
votes
2answers
282 views
Diophantine equation: getting a list of solutions for each coeficient
I'm trying to solve the following diophantine equation,$$x^3+dy^3=1$$
such that $$x,y \in \mathbb{Z} \land y \neq 0$$
and I want to form a list of solutions for each $$d\in[1,1000]; d\in\mathbb{N}$$
...
5
votes
2answers
256 views
Optimization problems over the Integers
Should be an easy question. How can I define a function which gets only integers in a form that I can use to find max and min for instance?
Example:
Find the Max of the expression ...
4
votes
0answers
270 views
Solving Thue equations
A Thue equation is a 2-variable homogeneous integer polynomial of degree at least 3. It's well-known that such equations have only finitely many solutions over the integers.
I'm trying to solve some ...
-1
votes
1answer
89 views
equidistant solutions in Sequence Sums That Are Squares
In my demonstration "Sequence Sums That Are Squares"
I demonstrate two "lines" of solutions (of infinite length) and ask whether someone can find one more line.
I am looking forward to your solution. ...
2
votes
2answers
139 views
Problem on Consecutive Sequence Sums that are Squares
Here is a seemingly simple Problem:
I have two natural numbers n and m, n < m such that
S1:= n +...+ m
is a square and also
S2:= n +...+ m + (m+1)
is a square.
Problem a) : Find n and m.
You ...
2
votes
4answers
3k views
find the number of integral solutions a+b+c+d+e+f = 18 [duplicate]
Find the number of integral solutions of
a + b + c + d + e + f = 18
where a, b, c, d, e, f are elements of the range ...
10
votes
2answers
433 views
Efficient way to solve equal sums $x_1^k+x_2^k+\dots+x_5^k=y_1^k+y_2^k+\dots+y_5^k$ with Mathematica?
I need to solve the system of equations, call it $S_1$, in the integers
$$x_1x_2x_3x_4x_5 = y_1y_2y_3y_4y_5$$
$$x_1^k+x_2^k+\dots+x_5^k=y_1^k+y_2^k+\dots+y_5^k,\;\; k= 2,4,6$$
I used a very ...
6
votes
1answer
2k views
How to solve this equation with integers as a solution?
I want to solve the equation $$x^y + y = y^x + x$$
with $x$, $y$ are integer numbers. I tried
Solve[x^y + y == y^x + x, {x, y}, Integers]
How to solve the ...
6
votes
3answers
317 views
How do I generate a set of n-tuples containing integral solutions to a linear equation provided certain constraints?
Let $m,k,p$ be fixed positive integers. I want to create a table of k-tuples $(x_1,x_2,\ldots,x_k)$ comprised of solutions in positive integers to the equation below:
$$x_1+x_2+\cdots+x_k=m\quad\...
5
votes
2answers
1k views
Extracting Reduce results
I'm solving a Diophantine equation inside of a function using Reduce but I'm having trouble extracting the necessary parts of the answer.
For example, if my input ...
