# Questions tagged [diophantine-equations]

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

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### Why doesn't FindInstance work in finding Heronian triangle?

I was trying to find a triangle whose side lengths are all positive integers and area is rational number (similar to Heronian triangle) with FindInstance, but it ...
• 1,683
1 vote
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### Finding integer solutions to $\sqrt{a+b^2+c^3}=a-b-c$ where $a,b,c\in\mathbb Z$ and $a\ne b\ne c$

So I want to use Mathematica to find integer solutions to the following Diophantine equation$$\sqrt{a+b^2+c^3}=a-b-c$$with$$a,b,c\in\mathbb Z\qquad a\gt b\gt c$$I have managed to find a solution on my ...
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### Does a(a+1)(a+2)-1=b²+2=10c+3 have solutions over natural numbers? If yes, how many?

It is a question from the math olympiad I was participating in that happened like a month ago. The provided solution turned out to be wrong. It isn't the question itself, but the solution basically ...
228 views

### Reduce and Solve fail to provide explicit integer solutions

Why is Reduce or Solve unable to provide explicit solutions over integers for such simple system of equations? ...
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1 vote
161 views

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### 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 ...
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### 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 ...
151 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
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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 ...
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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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### 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, ...
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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....
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### 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 $r\ge3$: $$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$ ...
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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 ...
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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] ...
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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 ...
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1 vote
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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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