Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
2
votes
2answers
112 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
74 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
724 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
85 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
37 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
72 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
208 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
633 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
$$\frac{1}{k_1}+\cdots+\frac{1}{...
-3
votes
2answers
278 views
Solving a diophantine equation with three variables
Solve this equation using Mathematica:
$2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.
6
votes
2answers
126 views
Unexpected behaviour with FindInstance
Evaluating
FindInstance[a/(b + c) + b/(a + c) + c/(a + b) == 4, {a, b, c}, Integers]
does what I'd expect it to do: find a solution to the equation in integers.
...
2
votes
1answer
95 views
Storing Multiple solutions from Solve as rows of a matrix
I am trying to find all the solutions to a Diophantine equation (only integer valued solutions) using Solve[], however there are multiple and even many solutions sometimes.
How can I store these ...
5
votes
1answer
330 views
Solving a Diophantine equation with a large solution
I am trying to solve the following Diophantine equation with Mathematica:
$\frac{x}{y+z}+\frac{z}{x+y}+\frac{y}{x+z}=4$
It is known that there are three positive numbers that satisfy the equation ...
2
votes
0answers
53 views
Introducing constraints in FrobeniusSolve
I have a knapsack problem. Say I have N possible items $x_i$. I would like to know:
$ \sum_{i=0}^{2} c_i x_i = W$
With the following constraint:
$c_i=3 \lor 4$
As an example suppose the set $x_i=...
2
votes
0answers
109 views
How to test solvability of a Diophantine equation?
When applying the BRC theorem I need to test if an equation has a non-zero solution in integers.
The equation is of form:
$$x^2 + by^2 + cz^2 = 0$$
We could try to directly check if this equation ...
2
votes
2answers
206 views
Integer partitions without repetitions
What combination of numbers makes a specific sum?
The code below is not very effective, because it also gives answers in which a number is used more than once even though it was given in the list ...
5
votes
2answers
115 views
Finding all solutions in the Roth's theorem
Roth's theorem. For all algebraic irrational $\alpha$
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$
with $\epsilon>0$, has finitely many ...
16
votes
5answers
2k views
Puzzle — 20 People to consume 20 units of food under constraints
I am learning Mathematica because I love it. I also love solving puzzles so I think it would be a nice way to learn Mathematica through puzzles. This is first puzzle in series I intend to solve.
So ...
0
votes
1answer
206 views
Solving Diophantine equations
Given a complex number $z$ and a positive integer $n$, I would like to be able to find integer solutions $\alpha,\beta,\gamma$ to the Diophantine equation
$$0 < a^2\vert z \vert^2 + \beta\textrm{ ...
4
votes
3answers
151 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
213 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
788 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
81 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
196 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
174 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
113 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
727 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
265 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
207 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
218 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
137 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
374 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
1k 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
267 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:
$$x_1+x_2+\cdots+x_k=m\quad\...
5
votes
2answers
908 views
Extracting Reduce results
I'm solving a Diophantine equation inside of a function using Reduce but I'm having trouble extracting the necessary parts of the answer.
For example, if my input ...
7
votes
0answers
445 views
Computing Ehrhart's polynomial for a convex polytope
Is there a Mathematica implementation for computing the Ehrhart polynomial of a convex polytope which is specified either by its vertices or by a set of inequalities?
I am interested in knowing this ...
14
votes
6answers
5k views
How to find lattice points on a line segment?
How do I find points on the line segment joining {-4, 11} and {16, -1} whose coordinates are positive integers?
10
votes
2answers
1k views
Solving/Reducing equations in $\mathbb{Z}/p\mathbb{Z}$
I was trying to find all the numbers $n$ for which $2^n=n\mod 10^k$ using Mathematica.
My first try:
Reduce[2^n == n, n, Modulus -> 100]
However, I receive ...
14
votes
2answers
603 views
Is there a way to use functions like Prime[n] within Solve[]?
I'm trying to see if a number can be written as the sum of two prime numbers. Ideally, I would like to use
Solve[ Prime[n] + Prime[m] == 100, {n, m}]
But that ...
3
votes
1answer
1k views
How to solve this equation with positive integers as a solutions?
This is a problem of United Kingdom Mathematical Olympiad. Find all triples $(x,y,z)$ of positive integers such that
$$\biggl(1+\dfrac{1}{x}\biggr)\cdot \biggl(1+\dfrac{1}{y}\biggr)\cdot \biggl(1+\...
1
vote
1answer
959 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:
$$LHS[1] \leq RHS[1]$$
$$LHS[2] \leq RHS[2]$$
$$LHS[3] \leq RHS[3]$$
$$\vdots$$
$$LHS[n]...
9
votes
3answers
3k views
Finding the number of solutions to a diophantine equation
I want to count total number of the natural solutions (different from 0) of the equation $2x + 3y + z = 100$, but don't know how. How can I calculate it using Mathematica? I tried:
...
8
votes
3answers
2k views
How can I solve the equation with integers as a solution?
I want to solve the equation $$(x-1)^2 + (y-1)^2 + (z-1)^2 = 49$$ where $x$, $y$, $z$ are integer and $x \neq 1$, $y \neq 1$, $x \neq 1$. How do I tell Mathematica to do that?
8
votes
4answers
709 views
Defining a Unique Domain for Solving Diophantine Equations
I am working on a research problem in discrete geometry to do with sphere packings, and believe it or not, I have been able to reduce it to finding the solutions to the Diophantine equation,
$$n = \...
11
votes
3answers
571 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.
...
