All Questions
Tagged with plane-geometry nt.number-theory
7 questions
1
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Show that a region in a plane defined by a polynomial contains integer points
Let $F \in \mathbb{Z}[x,y]$ be a polynomial of degree $2n$ such that the homogeneous degree $2n$ part of $F$, say $F_{2n}$, is positive semi-definite. How does one show that for some $\delta_n > 0$ ...
- 26.9k
6
votes
2
answers
544
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On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
- 5,979
15
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1
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Lines passing through many points of the form $(c^n,c^m)$
For $c>1$ consider the subset $X\subset \mathbb R^2$ consisting of all points $(c^n,c^m)$ where $n,m\in \mathbb Z$.
Question. Suppose $L\subset \mathbb R^2$ is a line that is not horizontal, not ...
- 14.3k
7
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Is $\arcsin(1/4) / \pi$ irrational?
Is $\arcsin(1/4) / \pi$ rational? An approximation given by a calculator seem to suggest that it isn't, but I found no proof. Thanks in advance!
- 3,274
7
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553
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Two queries on triangles, the sides of which have rational lengths
Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational.
We are aware that a positive integer is called "congruent" only if it is the area of a right ...
- 4,746
3
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Number of distinct directions in the set $\mathbb{Z}^2 \cap B(0,R)$
We say that non-zero directions $v, w \in \mathbb{R}^2$ are equivalent if they span the same line (i.e. $\exists C \in \mathbb{R}: v = Cw$.), and distinct otherwise. Given a collection $V \subset \...
- 81
17
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What is the smallest positive integer for which the congruent number problem is unsolved?
The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...
- 41.4k