9
votes
Accepted
7
votes
How can I select four points on a sphere to make a regular tetrahedron so that its coordinates are integer numbers?
EDIT: Improved distance computation efficiency a lot by using DistanceMatrix:
...
- 19.1k
7
votes
Accepted
How can I select four points on a sphere to make a regular tetrahedron so that its coordinates are integer numbers?
No such regular tetrahedron,even no triangle of such regular tetrahedron.
If there are exist such regular tetrahedron, the length of its side should be ...
- 72.9k
6
votes
Accepted
How can I get all squares on this sphere so that its coordinates are integer numbers?
For four points {p1,p2,p3,p4} on the sphere, if exist two segments bisect each other ...
- 72.9k
6
votes
Finding integer solutions to $\sqrt{a+b^2+c^3}=a-b-c$ where $a,b,c\in\mathbb Z$ and $a\ne b\ne c$
Note that, from the given Equation, we must have the condition $a - b - c \geqslant 0$. I consider $1\leqslant a,b,c\leqslant 100$.
...
- 3,505
6
votes
Finding integer solutions to $\sqrt{a+b^2+c^3}=a-b-c$ where $a,b,c\in\mathbb Z$ and $a\ne b\ne c$
Alternative without Sqrt
...
- 53.8k
6
votes
Accepted
Finding integer solutions to $\sqrt{a+b^2+c^3}=a-b-c$ where $a,b,c\in\mathbb Z$ and $a\ne b\ne c$
With changing the condition a!=b!=c to a>b>c we get:
Solve[Sqrt[a + b^2 + c^3] == a - b - c && a > b > c, {a, b,
c}, Integers]
And if we un-...
- 51.8k
6
votes
Accepted
Extremum of the graph of a function has integer coordinates
My answer is based on another answer on this site which I do not remember its link. I am sorry the author of this answer.
PS. My answer is based on this answer
I use your result
...
- 3,505
6
votes
Solving a-two-variable equation in primes
FindInstance cannot prove that there are no solutions, especially when the domain of solutions is the set of primes.
More acceptable approach would be:
...
- 57.3k
6
votes
How can I select four points on a sphere to make a regular tetrahedron so that its coordinates are integer numbers?
I used PowersRepresentations as Daniel Lichtblau did here as it felt natural to me when I first saw this type of problem.
With Version 3, I get the 130 solutions ...
- 3,728
5
votes
Accepted
5
votes
Accepted
Finding coprime solutions to Diophantine equations
Reduce[x^2 + y^2 == 17 && x > 0 && y > 0 && GCD[x, y]==1, {x, y}, Integers]
returns
...
- 12.1k
4
votes
Accepted
Solving a-two-variable equation in primes
Indeed, this can be done in one line:
FindInstance[x^3 - y^4 == 1, {x, y}, Primes]
{}
No solution in the primes.
- 26.4k
4
votes
Accepted
4
votes
Reduce and Solve fail to provide explicit integer solutions
Also works with some alternate variable orders. For instance:
...
- 2,777
4
votes
Reduce and Solve fail to provide explicit integer solutions
What is even more strange that using Reduce twice (or Reduce with Solve) provides the ...
- 16.9k
4
votes
How can I find four points on a circle to make a square so that its coordinates are integer numbers?
I have voted for @cvgmt. This is for fun:
The rational points of circle with denominator 25 can generated (excluding degenerate points):
...
- 60.8k
3
votes
How can I get all squares on this sphere so that its coordinates are integer numbers?
Use the Graph Theory method from https://mathematica.stackexchange.com/a/289130/72111 make it faster.
At first we search the space quadrilateral which four sides are equal by the graph theory method,...
- 72.9k
2
votes
Reduce and Solve fail to provide explicit integer solutions
Here is a partial workaround in 13.3.1 on Windows 10:
FindInstance[{x1 x3 == 1, x2 x3 + x1 x4 == 5, 2 x2 x4 == 12}, {x1, x2, x3, x4}, Integers, 5]
FindInstance::...
- 26.4k
2
votes
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?
No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and ...
- 16.9k
2
votes
How can I get all squares on this sphere so that its coordinates are integer numbers?
Definitely a task for PowersRepresentations.
First find triples of nonnegatives with squares summing to 15^2.
...
- 59k
1
vote
Parametric integer solution of $aX^2+bY^2-cZ^2=0$
Use Solve or Reduce like:
Reduce[{a x^2 + b y^2 - c z^2 == 0, {a, b, c} \[Element]
PositiveIntegers}, {x, y, z}, Reals]
You may plot the solution like:
...
- 51.8k
Only top scored, non community-wiki answers of a minimum length are eligible