Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
71
questions
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votes
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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:
...
- 199
1
vote
0
answers
77
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
131
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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0
votes
1
answer
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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 $$p=q^2∗r^2∗y^4∗z^2∗(2∗w^4+2∗r^2∗w^4∗y^4∗z^2+r^2∗w^8∗y^4∗z^2+2) + q^2+ 2∗r^6∗w^4∗y^m∗z^6∗(w+w^2+...
- 121
1
vote
0
answers
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views
Why does `FindInstance` refuse to find a second instance, in a number theory problem?
Hi I am looking for reversible Markov generating matrices of order three, whose upper (2,2) block has integer eigenvalues. There are 5 free parameters; I specified 2
...
- 896
2
votes
1
answer
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views
Strange results from FrobeniusSolve
As part of pruning for my code searching for optimal addition chains I want to try and find some fast ways to discover that certain numbers are not representable in the Frobenius coin problem.
An ...
- 123
3
votes
2
answers
116
views
Diophantine inequality that's not solved
Mathematica has trouble solving this Diophantine inequality:
...
- 416
0
votes
1
answer
46
views
How do I find the solution of a system of Diophantine equations in two variables? [closed]
If the first equation is 2x+y=152, and the second is x+2y=100, and it is a fact that both variables are whole numbers and x is larger than y, what is the value of x?
- 11
2
votes
2
answers
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Solving a diophantine equation in 'large' values
Let's first discuss what I am trying to solve: I want to solve the diophantine equation stated below for relatively 'large' values of $r$.
$$\frac{a(a + 3)(a(r - 9) + (7 - r))}{12}=\frac{b (3 + b (-5 +...
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0
votes
1
answer
479
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
450
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 $r\ge3$:
$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$
...
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2
votes
1
answer
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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
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votes
1
answer
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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]
...
- 2,145
0
votes
2
answers
107
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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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1
vote
1
answer
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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:
$n\ne a\ne b\ne c\ne d\ne k\ne f$;...
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3
votes
1
answer
260
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, ...
- 1,801
4
votes
2
answers
204
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 ...
- 15.7k
8
votes
3
answers
368
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 ...
- 1,723
1
vote
1
answer
111
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:
$$\varphi\left(\text{a},\text{b},\text{c}\...
- 1,723
2
votes
2
answers
149
views
Finding a program that can find integer solutions for large values of a variable
I've the following code:
...
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2
votes
0
answers
65
views
Never terminating when solving linear Diophantine systems
Say I have:
$(1):$ Set of Diophantine systems where each defines a (possibly unbounded) polyhedron.
This is solvable with existing algorithms:
$(*):$ The $(1)$ can be solved in SageMath with ...
- 612
-1
votes
1
answer
69
views
Quadratic equation with integral coefficients [closed]
Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2+bx+c=0$ are distinct and both lie in the interval
(0,1)
(1,2)
(2,3)
(Brackets signify open interval, roots are $IN BETWEEN $ ...
4
votes
1
answer
889
views
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Why does Solve lock up when trying to solve the equation
...
- 43
2
votes
4
answers
267
views
FrobeniusSolve with solutions only being 0 or 1 being acceptable
I want an EFFICIENT way of solving the Frobenius Equation
a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U
where n1, n2, ..., nK are restricted to be either 0 or 1. I ...
- 1,398
3
votes
4
answers
207
views
How do I declare a countably infinite list of variables as being integers?
I want to do this,
Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100},
{n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ]
...
- 1,398
2
votes
2
answers
133
views
On the solvability of the Diophantine equation $a(x^2-y^2)+2bxy=1$
Take two integers $a,b$ with $\mathrm{GCD}(a,b)=1$ and $a$ odd. Consider the Diophantine equation
$$
a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z
$$
Is there any efficient way to determine whether such ...
0
votes
1
answer
132
views
Unexpected omission by Wolfram Alpha [closed]
When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2
- 111
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votes
2
answers
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views
How can I reduce the time to run this code?
I am trying to find the integers $ a, b, c, d \in [-15, -1] \cup [1, 15] $ so that the equation $ \left| x^2 + a x + b \right| = c x + d $ has four distinct integeral solutions different from $ 0 $.
...
- 1,801
3
votes
1
answer
104
views
Guess Diophantine equation from its solutions
Take for example the equation
$$
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
$$
Can we ...
1
vote
1
answer
294
views
How do we show reduction steps?
This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper.
...
- 2,288
4
votes
3
answers
529
views
Finding Smallest Positive Solution to Diophantine Equation
Mathematica can solve $v^2- d u^2=4$ quickly, even for nonsmall $d$:
d = 400004;
Reduce[v^2 - d u^2 == 4, Integers]
d = 400012;
Reduce[v^2 - d u^2 == 4, Integers]
...
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3
votes
3
answers
680
views
Solving a system of diophantine equations from a mathematical competition
There is a problem from the $66th$ Putnam Mathematical Competition, $B2$
Find all positive integers $n,k_1,k_2,...,k_n$ such that $$k_1+\cdots+k_n=5n-4$$
and
$$\frac{1}{k_1}+\cdots+\frac{1}{...
- 8,648
-3
votes
2
answers
409
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
203
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
203
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
550
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 ...
- 798
5
votes
1
answer
601
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
74
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=...
- 2,058
2
votes
0
answers
196
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
315
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
148
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
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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
657
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
$$0 < a^2\vert z \vert^2 + \beta\textrm{ ...
- 119
4
votes
3
answers
367
views
find and count the number of solutions without multiplicity in Solve?
I would like to solve a Diophantine equation and find its solution, but I need only count one time for each $a$, i.e., when for some $a$ it found some $x,y,z$, then go to the next $a$. more precisely ...
- 798
2
votes
2
answers
334
views
Solve an exponential equation in integers
Find two integers x and y such that $x^y +y^x = 94032$.
I have used
...
- 2,145
20
votes
3
answers
975
views
Partitioning a number into consecutive integers
Consider n=45; then
$$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$$
Question: how to find all representations of a ...
- 23.2k
1
vote
0
answers
97
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diophantine linear equation with condition on the sum of coefficients [closed]
I am trying to solve the following diophantine equation
x*a + y*b == c
where a,b,c are integers and the absolute value of their sums is, for instance, 4. Then I ...
2
votes
2
answers
381
views
how to find sum of variable c1+c2+c3 for expression combinations
how to find sum of variable c1 + c2 + c3 for expression combinations:
...
- 21
5
votes
3
answers
227
views
On Solutions of Diophantine Equation
By Mathematica, I try to obtain solutions of Diophantine equations such as:
$$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 &...
- 729
6
votes
1
answer
134
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:
...
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