Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
92
questions
1
vote
3
answers
145
views
Finding integer solutions to $\sqrt{a+b^2+c^3}=a-b-c$ where $a,b,c\in\mathbb Z$ and $a\ne b\ne c$
So I want to use Mathematica to find integer solutions to the following Diophantine equation$$\sqrt{a+b^2+c^3}=a-b-c$$with$$a,b,c\in\mathbb Z\qquad a\gt b\gt c$$I have managed to find a solution on my ...
- 233
1
vote
0
answers
117
views
Does a(a+1)(a+2)-1=b²+2=10c+3 have solutions over natural numbers? If yes, how many?
It is a question from the math olympiad I was participating in that happened like a month ago. The provided solution turned out to be wrong.
It isn't the question itself, but the solution basically ...
3
votes
3
answers
226
views
Reduce and Solve fail to provide explicit integer solutions
Why is Reduce or Solve unable to provide explicit solutions over integers for such simple system of equations?
...
- 16.9k
1
vote
2
answers
160
views
How can I choose the integer numbers a, b, c, d, t so that the equation $\sqrt{a x^2+b x+c}=d x+t$ has two integer solutions?
The equation $\sqrt{x^2+36 x+180}=2 x+15$ have two integer solutions are $x = -5$ and $x = -3$. How can I choose the integer numbers $a, b, c, d, t$ so that the equation $\sqrt{a x^2 + b x + c} = d x +...
- 25
1
vote
0
answers
80
views
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 ...
- 3,728
0
votes
1
answer
73
views
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 $x=1,y=4$...
- 247
3
votes
1
answer
176
views
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 ...
- 1,032
1
vote
2
answers
112
views
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
2
votes
2
answers
368
views
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 ...
- 1,315
10
votes
3
answers
905
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 ...
- 1,032
9
votes
3
answers
417
views
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 ...
- 1,032
5
votes
2
answers
227
views
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
...
- 1,032
1
vote
1
answer
105
views
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
3
votes
2
answers
307
views
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.
...
- 1,315
20
votes
4
answers
969
views
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 ...
- 73.1k
2
votes
4
answers
287
views
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 ...
- 259
2
votes
1
answer
193
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
619
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
...
- 3,515
2
votes
0
answers
91
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
148
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
89
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
197
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 ...
- 41.2k
5
votes
2
answers
567
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, ...
- 41.2k
0
votes
1
answer
71
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 $$p=q^2∗r^2∗y^4∗z^2∗(2∗w^4+2∗r^2∗w^4∗y^4∗z^2+r^2∗w^8∗y^4∗z^2+2) + q^2+ 2∗r^6∗w^4∗y^m∗z^6∗(w+w^2+...
- 121
1
vote
0
answers
61
views
Why does `FindInstance` refuse to find a second instance, in a number theory problem?
Hi I am looking for reversible Markov generating matrices of order three, whose upper (2,2) block has integer eigenvalues. There are 5 free parameters; I specified 2
...
- 1,798
2
votes
1
answer
79
views
Strange results from FrobeniusSolve
As part of pruning for my code searching for optimal addition chains I want to try and find some fast ways to discover that certain numbers are not representable in the Frobenius coin problem.
An ...
- 153
3
votes
2
answers
165
views
Diophantine inequality that's not solved
Mathematica has trouble solving this Diophantine inequality:
...
- 438
0
votes
1
answer
56
views
How do I find the solution of a system of Diophantine equations in two variables? [closed]
If the first equation is 2x+y=152, and the second is x+2y=100, and it is a fact that both variables are whole numbers and x is larger than y, what is the value of x?
- 11
2
votes
2
answers
128
views
Solving a diophantine equation in 'large' values
Let's first discuss what I am trying to solve: I want to solve the diophantine equation stated below for relatively 'large' values of $r$.
$$\frac{a(a + 3)(a(r - 9) + (7 - r))}{12}=\frac{b (3 + b (-5 +...
- 2,001
0
votes
1
answer
1k
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
634
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 $r\ge3$:
$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$
...
- 2,001
2
votes
1
answer
88
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 ...
- 145
0
votes
1
answer
57
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]
...
- 2,263
0
votes
2
answers
115
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 ...
- 2,001
1
vote
1
answer
83
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:
$n\ne a\ne b\ne c\ne d\ne k\ne f$;...
- 2,001
3
votes
1
answer
299
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, ...
- 3,515
4
votes
2
answers
252
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 ...
- 18.3k
9
votes
3
answers
536
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 ...
- 2,001
1
vote
1
answer
150
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:
$$\varphi\left(\text{a},\text{b},\text{c}\...
- 2,001
2
votes
2
answers
157
views
Finding a program that can find integer solutions for large values of a variable
I've the following code:
...
- 2,001
2
votes
0
answers
77
views
Never terminating when solving linear Diophantine systems
Say I have:
$(1):$ Set of Diophantine systems where each defines a (possibly unbounded) polyhedron.
This is solvable with existing algorithms:
$(*):$ The $(1)$ can be solved in SageMath with ...
- 622
-1
votes
1
answer
81
views
Quadratic equation with integral coefficients [closed]
Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2+bx+c=0$ are distinct and both lie in the interval
(0,1)
(1,2)
(2,3)
(Brackets signify open interval, roots are $IN BETWEEN $ ...
4
votes
1
answer
901
views
Why does Solve lock up when trying to solve the quadratic equation with large integers?
Why does Solve lock up when trying to solve the equation
...
- 43
2
votes
4
answers
327
views
FrobeniusSolve with solutions only being 0 or 1 being acceptable
I want an EFFICIENT way of solving the Frobenius Equation
a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U
where n1, n2, ..., nK are restricted to be either 0 or 1. I ...
- 1,678
3
votes
4
answers
263
views
How do I declare a countably infinite list of variables as being integers?
I want to do this,
Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100},
{n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ]
...
- 1,678
2
votes
2
answers
142
views
On the solvability of the Diophantine equation $a(x^2-y^2)+2bxy=1$
Take two integers $a,b$ with $\mathrm{GCD}(a,b)=1$ and $a$ odd. Consider the Diophantine equation
$$
a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z
$$
Is there any efficient way to determine whether such ...
0
votes
1
answer
141
views
Unexpected omission by Wolfram Alpha [closed]
When Alpha is submitted the equation $a(a^2-1)=2b^2$, it unexpectedly forgets the integer solution $a=1,b=0$. What could explain this ?
http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2
- 111
5
votes
2
answers
803
views
How can I reduce the time to run this code?
I am trying to find the integers $ a, b, c, d \in [-15, -1] \cup [1, 15] $ so that the equation $ \left| x^2 + a x + b \right| = c x + d $ has four distinct integeral solutions different from $ 0 $.
...
- 3,515
5
votes
1
answer
341
views
Pythagorean quadruples
I am very interested in the problem that deals with the Pythagorean quadrruples, which are listed in rosettacode. Unfortunately, I have not been able to find any way to compute them with Mathematica. ...
- 1,289
4
votes
1
answer
117
views
Guess Diophantine equation from its solutions
Take for example the equation
$$
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 ...