Questions tagged [diophantine-equations]

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

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2 votes
1 answer
149 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^...
4 votes
3 answers
441 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 ...
1 vote
0 answers
66 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
136 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: ...
1 vote
0 answers
81 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 ...
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0 votes
2 answers
136 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 ...
5 votes
2 answers
468 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, ...
0 votes
1 answer
41 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+...
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1 vote
0 answers
57 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 ...
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2 votes
1 answer
65 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 ...
3 votes
2 answers
123 views

Diophantine inequality that's not solved

Mathematica has trouble solving this Diophantine inequality: ...
  • 416
0 votes
1 answer
48 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?
2 votes
2 answers
106 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 +...
  • 1,765
0 votes
1 answer
632 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....
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3 votes
4 answers
489 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 $r\ge3$: $$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$ ...
  • 1,765
2 votes
1 answer
85 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 ...
  • 135
0 votes
1 answer
49 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] ...
  • 2,175
0 votes
2 answers
107 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 ...
  • 1,765
1 vote
1 answer
78 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: $n\ne a\ne b\ne c\ne d\ne k\ne f$;...
  • 1,765
3 votes
1 answer
270 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 votes
2 answers
212 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 ...
  • 16.3k
8 votes
3 answers
387 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 ...
  • 1,765
1 vote
1 answer
118 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}\...
  • 1,765
2 votes
2 answers
153 views

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

I've the following code: ...
  • 1,765
2 votes
0 answers
67 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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-1 votes
1 answer
70 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
895 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
4 answers
276 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
4 answers
218 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
2 answers
135 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
132 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
2 answers
769 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
1 answer
105 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 ...
1 vote
1 answer
315 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
3 answers
566 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] ...
4 votes
3 answers
714 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}{...
  • 8,668
-3 votes
2 answers
417 views

Solving a diophantine equation with three variables

Solve this equation using Mathematica: $2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.
7 votes
2 answers
208 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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2 votes
1 answer
224 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 ...
1 vote
2 answers
555 views

determine the coefficients of a quadratic equation with "Solve"

Assume we want to determine the coefficients (integer) of a quadratic equation by having some information about the input and output values. If I enter the input and output individually before the ...
  • 808
5 votes
1 answer
619 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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2 votes
0 answers
78 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=...
  • 2,078
2 votes
0 answers
209 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 ...
  • 325
2 votes
2 answers
325 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
2 answers
149 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 ...
  • 8,668
16 votes
5 answers
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
1 answer
709 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
3 answers
391 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 ...
  • 808
2 votes
2 answers
338 views

Solve an exponential equation in integers

Find two integers x and y such that $x^y +y^x = 94032$. I have used ...
  • 2,175
21 votes
3 answers
991 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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