**Geometry/Number Theory Open Problems**

Index of Problems

### Points on a Parabola

**RESEARCHER:** Nate Dean

**OFFICE: **Chair of Mathematics, Texas Southern University

**Email:**dean_nx@tsu.edu

**DESCRIPTION:**The problems is this: How many points can you find
on the (half) parabola y=x^2, x>0, so that the distance between any
pair of them is rational? This problem sounds like a geometry
problem, but it is likely to require techniques in number theory.
That's because, to determine if the distance between (a,b) and (s,t)
is rational, you need to test whether (a-s)^2 + (b-t)^2 is the square
of a rational number...and such questions typically fall into the
realm of number theory.
Of course, since this is an open problem, no-one can claim to know just
what field the problem lies in, since no-one knows the solution.

I believe it is not even known if there are more than 5 points on the
parabola which satisfy the condition. It is certainly not known if
there is an infinite family with pairwise rational distances. We can
quickly see that there are three such points in the following way:
Pick two rational numbers, say 1/5 and 3/5, and let those be the
distances between the pairs of points AB and BC. If you fix those
distances and slide the points up the parabola, the distance AC will
gradually increase, bounded above by 4/5. Since the rational numbers
are dense in the reals, there will be many placements of the points so
that AC is a rational distance.

It is not too hard to show that if you have N points on the parabola
with rational distances between them, then you can find N points on
the parabola with *integer* distances between them.

### Chromatic Number of the Plane

**RESEARCHER:** Robert Hochberg

**OFFICE: **CoRE 414

**Email:**hochberg@dimacs.rutgers.edu

**DESCRIPTION:** Assign a color to each point of the plane so that
pairs of points whose distance is exactly 1 get different colors. The
minimum number of colors required to achieve this is called the
"chromatic number of the plane," and noone knows what it is.
This problem is closely related to the problem of finding unit distance graphs with
large chromatic number. This problem has been open since it was
first posed by Edward Nelson in 1956. At that time, it was proved
that the chromatic number of the plane was either 4, 5, 6 or 7, and no
improvements on those bounds have been made since! The coloring of
the plane shown to the right, where each square has side 3/5, shows
that the chromatic number of the
plane is at most 9. Can you find a coloring with 7 colors to get the
current best-known upper bound?

This problem is *very* hard, judging by the length of time that
has gone by with no improvements on the bounds. It will probably take
some brand new idea to make a contribution. So, what are you waiting
for?!

**Back to main page**