Given the series of prime numbers greater than 9, we can organize them in four rows, according to their last digit ($1,3,7$ or $9$).
The column in which they are displayed is the ten to which they belong, as illustrated in the following scheme.
The blue points represent the primes, whereas the red points represent the integers that are located in the rows $1,3,7,9$ but that are not primes.
For instance, in correspondence of the second ten ($x$-axis) we find two red points in the rows $1$ and $7$ ($y$-axis), because $20+1=21$ and $20+7=27$ are not primes.
I have conjectured that given any two red points, it is always possible to find an ellipse with foci in these two points and passing through at least one blue point (a prime number) and at least another red point (a composite number).
Here I show some examples:
Similarly, it can be conjectured that given any two blue points, it is always possible to find an ellipse with foci in these points and passing through at least one prime and at least one composite. Here follows some examples:
My question is:
If true, is this property related to the distribution of the prime numbers? Or it is only due to the particular reference system (lattice) I used to represent them?
Thanks for your suggestions and comments, and sorry if the whole problem may be naive.