Questions tagged [diophantine-equations]

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

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40 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] ...
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0answers
60 views

7X7 matrix of distinct cubed integers with identical row and column sums

Problem: find a 7X7 matrix of distinct cubed integers with identical row and column sums. Well, I have the following code (that came from @kglr from my previous question): ...
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2answers
88 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 ...
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1answer
70 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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1answer
205 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, ...
4
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2answers
187 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 ...
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3answers
236 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 ...
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1answer
80 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}\...
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2answers
131 views

Finding a program that can find integer solutions for large values of a variable

I've the following code: ...
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0answers
59 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 ...
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58 views

LinearProgramming with constraint on result

I need to solve a bunch of Diophantine Equations that takes so looong with FindInstance/Reduce, so I tried LinearProgramming and it's very fast. Here is a simple one: ...
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1answer
56 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 $ ...
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1answer
879 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 ...
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4answers
236 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 ...
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4answers
181 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} ] ...
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2answers
125 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 ...
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1answer
123 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
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2answers
752 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
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1answer
101 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 ...
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1answer
145 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. ...
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3answers
386 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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3answers
660 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}{...
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2answers
352 views

Solving a diophantine equation with three variables

Solve this equation using Mathematica: $2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.
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2answers
178 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. ...
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1answer
154 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 ...
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1answer
496 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 ...
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0answers
68 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=...
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0answers
153 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 ...
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2answers
267 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 ...
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2answers
131 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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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 ...
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1answer
447 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
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3answers
245 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 ...
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2answers
275 views

Solve an exponential equation in integers

Find two integers x and y such that $x^y +y^x = 94032$. I have used ...
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3answers
892 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 ...
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0answers
92 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 ...
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2answers
330 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: ...
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3answers
198 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 &...
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1answer
118 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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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
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2answers
282 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 $$d\in[1,1000]; d\in\mathbb{N}$$ ...
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2answers
256 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 ...
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0answers
270 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 ...
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1answer
89 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. ...
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2answers
139 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
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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
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2answers
433 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
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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
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3answers
317 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
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2answers
1k 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 ...