3. Postdoctoral Program - MSRI

2.5 Non-Concyclic Points on a Parabola
Since the case for nonconcyclic points on a parabola is more difficult, we want
to ensure that the rational distance set of four rational points on the parabola is
nonconcyclic. We therefore need to examine the last condition of Theorem (4):
g(m13) + g(m24) = g(m23) + g(m14) = g(m12) + g(m34), in order to find our six
mij values. Consider the surface g(mij) + g(mkl) = t. We can make the following
substitutions:
mij = X2 + X
Y − T X , mkl =
X + 1
Y − T X
T − 2b
, t = ,
a
to obtain the elliptic curve E : Y 2 = X 3 + (T 2 + 2)X 2 + X. Thus if we find ratio-
nal points on the elliptic curve we can find rational mij values that satisfy the last
condition.
2.6 Example
In order to illustrate this process, we include the following example:
Given the parabola y = x 2 , let T = 13
6
E : Y 2 = X3 + 241
36 X2 + X. We chose T = 13
6
gives us an elliptic curve of positive rank.
in order to get the following elliptic curve
because it is a small rational value that
We get the rational points Q1 = (6: 19: 18), Q2 = (3: 13: 36), Q3 = (30: 169: 750) on
E. Now our mij have the following values: m12 = 3/10, m13 = 1/2, m23 = 4/3, m14 =
4, m24 = 6, m34 = 15/2. Which yields the following rational distance set:
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