Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
4
votes
1answer
859 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
175 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
162 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
114 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
1answer
84 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
728 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
92 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 ...
0
votes
0answers
43 views
When can FindInstance prove no solutions for a nonlinear Diophantine equation?
A few months ago, I thought of a marvelous proof that $a^3+b^3=2c^3$ has no nontrivial integer solutions in $\mathbb{N}$, but the margin of my cocktail napkin was too small to contain it. Now I've ...
1
vote
1answer
78 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
239 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
638 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
$$\frac{1}{k_1}+\cdots+\frac{1}{...
-3
votes
2answers
300 views
Solving a diophantine equation with three variables
Solve this equation using Mathematica:
$2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.
6
votes
2answers
137 views
Unexpected behaviour with FindInstance
Evaluating
FindInstance[a/(b + c) + b/(a + c) + c/(a + b) == 4, {a, b, c}, Integers]
does what I'd expect it to do: find a solution to the equation in integers.
...
2
votes
1answer
112 views
Storing Multiple solutions from Solve as rows of a matrix
I am trying to find all the solutions to a Diophantine equation (only integer valued solutions) using Solve[], however there are multiple and even many solutions sometimes.
How can I store these ...
5
votes
1answer
365 views
Solving a Diophantine equation with a large solution
I am trying to solve the following Diophantine equation with Mathematica:
$\frac{x}{y+z}+\frac{z}{x+y}+\frac{y}{x+z}=4$
It is known that there are three positive numbers that satisfy the equation ...
2
votes
0answers
57 views
Introducing constraints in FrobeniusSolve
I have a knapsack problem. Say I have N possible items $x_i$. I would like to know:
$ \sum_{i=0}^{2} c_i x_i = W$
With the following constraint:
$c_i=3 \lor 4$
As an example suppose the set $x_i=...
2
votes
0answers
113 views
How to test solvability of a Diophantine equation?
When applying the BRC theorem I need to test if an equation has a non-zero solution in integers.
The equation is of form:
$$x^2 + by^2 + cz^2 = 0$$
We could try to directly check if this equation ...
2
votes
2answers
215 views
Integer partitions without repetitions
What combination of numbers makes a specific sum?
The code below is not very effective, because it also gives answers in which a number is used more than once even though it was given in the list ...
5
votes
2answers
118 views
Finding all solutions in the Roth's theorem
Roth's theorem. For all algebraic irrational $\alpha$
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$
with $\epsilon>0$, has finitely many ...
16
votes
5answers
2k views
Puzzle — 20 People to consume 20 units of food under constraints
I am learning Mathematica because I love it. I also love solving puzzles so I think it would be a nice way to learn Mathematica through puzzles. This is first puzzle in series I intend to solve.
So ...
0
votes
1answer
246 views
Solving Diophantine equations
Given a complex number $z$ and a positive integer $n$, I would like to be able to find integer solutions $\alpha,\beta,\gamma$ to the Diophantine equation
$$0 < a^2\vert z \vert^2 + \beta\textrm{ ...
4
votes
3answers
163 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
231 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
815 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
83 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
233 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
178 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
115 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
268 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
223 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
227 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
87 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
138 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
390 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
272 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:
$$x_1+x_2+\cdots+x_k=m\quad\...
5
votes
2answers
939 views
Extracting Reduce results
I'm solving a Diophantine equation inside of a function using Reduce but I'm having trouble extracting the necessary parts of the answer.
For example, if my input ...
7
votes
0answers
457 views
Computing Ehrhart's polynomial for a convex polytope
Is there a Mathematica implementation for computing the Ehrhart polynomial of a convex polytope which is specified either by its vertices or by a set of inequalities?
I am interested in knowing this ...
14
votes
6answers
5k views
How to find lattice points on a line segment?
How do I find points on the line segment joining {-4, 11} and {16, -1} whose coordinates are positive integers?
10
votes
2answers
1k views
Solving/Reducing equations in $\mathbb{Z}/p\mathbb{Z}$
I was trying to find all the numbers $n$ for which $2^n=n\mod 10^k$ using Mathematica.
My first try:
Reduce[2^n == n, n, Modulus -> 100]
However, I receive ...
14
votes
2answers
624 views
Is there a way to use functions like Prime[n] within Solve[]?
I'm trying to see if a number can be written as the sum of two prime numbers. Ideally, I would like to use
Solve[ Prime[n] + Prime[m] == 100, {n, m}]
But that ...
3
votes
1answer
2k views
How to solve this equation with positive integers as a solutions?
This is a problem of United Kingdom Mathematical Olympiad. Find all triples $(x,y,z)$ of positive integers such that
$$\biggl(1+\dfrac{1}{x}\biggr)\cdot \biggl(1+\dfrac{1}{y}\biggr)\cdot \biggl(1+\...
1
vote
1answer
976 views
Using 'Reduce' to solve a set of inequalities, specified by a list
I have a two lists $LHS$ and $RHS$, both of size $n$. I want to solve a system of inequalities of the form:
$$LHS[1] \leq RHS[1]$$
$$LHS[2] \leq RHS[2]$$
$$LHS[3] \leq RHS[3]$$
$$\vdots$$
$$LHS[n]...
9
votes
3answers
3k views
Finding the number of solutions to a diophantine equation
I want to count total number of the natural solutions (different from 0) of the equation $2x + 3y + z = 100$, but don't know how. How can I calculate it using Mathematica? I tried:
...
8
votes
3answers
3k views
How can I solve the equation with integers as a solution?
I want to solve the equation $$(x-1)^2 + (y-1)^2 + (z-1)^2 = 49$$ where $x$, $y$, $z$ are integer and $x \neq 1$, $y \neq 1$, $x \neq 1$. How do I tell Mathematica to do that?
8
votes
4answers
720 views
Defining a Unique Domain for Solving Diophantine Equations
I am working on a research problem in discrete geometry to do with sphere packings, and believe it or not, I have been able to reduce it to finding the solutions to the Diophantine equation,
$$n = \...
11
votes
3answers
582 views
FindInstance with a Diophantine equation seems to go on forever
I tried to find a non-trivial integer solution to the equation
$$2012^2=a^2+b^2+c^2+d^2+e^2$$
with Mathematica but the computation takes minutes; I might be doing something wrong.
...
