# Questions tagged [diophantine-equations]

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

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### 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 ...
2answers
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### 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 ...
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### 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....
4answers
408 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 $r\ge3$: $$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$ ...
1answer
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### 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 ...
1answer
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### 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] ...
2answers
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### 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 ...
1answer
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### 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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### 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 ...
2answers
302 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 ...
2answers
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### 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 ...
5answers
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### 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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### 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: ...
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) ...