Questions tagged [diophantine-equations]
Questions on the use of Mathematica to find integer/rational solutions to equations.
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Finding integer solutions to an equation involving constrained integer constants
I have an equation involving two variables $x,y$ and 4 integer constants $e,f,b,c$. The constants also have a constraint on them. So something like $e+f-b+c=0$. I would like to find some integer ...
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Integer points on a line segment
I have two rational points. They are denoted below in black and they define a line segment. Find integer points on this segment.
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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 ...
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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 ...
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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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How can I choose the integer numbers a, b, c, d, t so that the equation $\sqrt{a x^2+b x+c}=d x+t$ has two integer solutions?
The equation $\sqrt{x^2+36 x+180}=2 x+15$ have two integer solutions are $x = -5$ and $x = -3$. How can I choose the integer numbers $a, b, c, d, t$ so that the equation $\sqrt{a x^2 + b x + c} = d x +...
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Solving Diophantine Equation with Only Specific Variable Values Allowed
I am trying to Solve an equation like this:
$$
\sum _{i=1}^{\text{iMax}} (-i+\text{iMax}+1) n(i)=546
$$
Where all the $n(i)$ are can only take on the values 2,3,6,7, or 8.
I really have no idea how to ...
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Finding coprime solutions to Diophantine equations
I'm hoping to use Mathematica to find solutions to Diophantine equations. Below is a toy example of something I would like to try.
Consider the case of $$x^2+y^2 = 17$$
which has the solution $x=1,y=4$...
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Extremum of the graph of a function has integer coordinates
Graph of the fuction $y=\dfrac{(x-26)(x+9)}{(x+14)(x+19)}$ has maximum point and minimum point are (-16,-49) and (-4,-1) whose ...
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Solving a-two-variable equation in primes
How solve the following equation in Mathematica (preferably in one line) for pairs of $(x,y)$ such that $x$ and $y$ are primes?
$x^3-y^4=1$
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How can I find all integer numbers so that mydistance is an integer number?
I am tring to selec two points A, B on the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 ==9^2 so that EuclideanDistance[pA,pB] is an ...
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How can I select four points on a sphere to make a regular tetrahedron so that its coordinates are integer numbers?
I want to select four points lie on the sphere (x-1)^2 + (y-3)^2 + (z-5)^2 = (5* Sqrt[3])^2 so that its coordinates are integer numbers to make a regular ...
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How can I get all squares on this sphere so that its coordinates are integer numbers?
I have the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 = 15^2. I want to select all squares on this sphere so that its coordiantes are twelve different integer numbers like ...
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How can I find four points on a circle to make a square so that its coordinates are integer numbers?
I have a circle and I want to select four points to make a square so that its coordiantes are eight different integer numbers like this
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