Given the series of prime numbers greater than $9$, we 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.
My conjecture is:
Given any two primes, it is always possible to find an ellipse whose foci coincide with the two points corresponding to the given primes in the previous representation, and passing through at least other two points, corresponding to other two primes.
Here I present some examples, where the red segments connect the two foci of each illustrative ellipse. Sorry if the picture is a bit chaotic!
Since I am not an expert of prime numbers, this can be an obvious result. In this case, I apologize for the trivial question. Anyway, I tried to prove this conjecture by means of the interesting observations related to this post, which is strongly related.
Thanks for your comments or suggestions, also to improve the quality and correctness of this question!
We just need to prove that there are two points such that the sum of each of their distances from the two focii is the same. The most obvious pair are those symmetrical with respect to the line linking the two focii, and/or its perpendicular bisector. For example, taking the primes $(3,7)$ and $(4,3)$, the primes $(1,3)$ and $(6,7)$ satisfy this condition.
The intuition is that eventually, given the infinite number of primes, one will always be able to find such pairs of numbers so that this condition is satisfied. However, I am unable to prove this, so I doubt that it is true for large primes, as prime gaps mean that it becomes increasingly unlikely that these numbers can be found.