Questions tagged [diophantine-equations]

Questions on the use of Mathematica to find integer/rational solutions to equations.

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2
votes
4answers
190 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
165 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} ] ...
4
votes
1answer
868 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 ...
0
votes
1answer
87 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
2
votes
2answers
119 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 ...
4
votes
2answers
732 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
95 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 ...
4
votes
3answers
268 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] ...
1
vote
1answer
81 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. ...
10
votes
2answers
424 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 ...
3
votes
3answers
642 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}{...
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: ...
5
votes
2answers
121 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 ...
-3
votes
2answers
310 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
145 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. ...
5
votes
1answer
384 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
1answer
121 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 ...
2
votes
0answers
60 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
119 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
224 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 ...
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
276 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
176 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
237 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
827 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 ...
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?
1
vote
0answers
84 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
257 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
184 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
3answers
281 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\...
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) ...
11
votes
3answers
586 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. ...
3
votes
2answers
273 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
231 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
238 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 ...
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?
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 ...
14
votes
2answers
629 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 ...
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 ...
7
votes
0answers
463 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 ...
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 ...
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: ...
5
votes
2answers
957 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 ...
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
990 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]...
8
votes
4answers
723 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 = \...