on the uniqueness of special-prime gaps I was studying a particular type of prime numbers, and I noticed an interesting property which I wish to prove or disprove.
Consider the set $S = \{10p + 1, 10p + 3, 10p + 7, 10p + 9\}$ (where p is a prime.)
If for any $i$ in S, $i$ is prime, then $i$ is a 'special-prime'. Now, with the help of python, we collected all special-primes $\le10^7$. Computing the differences between successive special-primes, I found that every difference is unique $\le10^7$.
More formally : let $x_i$ denote the ith special-prime. Then for all $i\le10^7$,
$x_{(i+1)} - x_i$ is unique. If I conjecture that this holds for all i, can you help me prove/disprove it.
 A: If special prime is defined to be the same as http://oeis.org/A227919, "primes which remain prime when rightmost digit is removed" then the originally posted answer below, about a different interpretation, applies with some obvious modifications. The triple $(p,10p+1,10p+19)$ is conjectured to be prime infinitely often, and most of those prime triples have no other primes between $10p+1$ and $10p+19$, so that difference $d=18$ would occur infinitely often.  Any possible $d$ would repeat at a frequency of $C_d\frac{x}{(\log x)^3}$ up to $x$.
The following interprets the question as asking about clusters $(10p+1,10p+3,10p+7,10p+9)$ all of whose elements are required to be prime (and $p$ also prime), with any element of the cluster designated as "special". 

The prime $k$-tuplets conjectures predict that all differences $d>4$ that occur are repeated infinitely often at a rate $C_d \frac{x}{(\log x)^6}$ up to $x$, where $C_d$ is a constant computable from $d$.
Differences of $2$ and $4$ are part of the definition of "special prime" quadruples and occur (according to the same conjectures) infinitely often.
The existence of difference $d$ between consecutive $x_i$ implies that 


*

*you can consistently add another requirement that $10p + A$ be prime, where $A$ would cause a difference of $d$ between two $x_i$ values (not necessarily consecutive)

*the conjectures say that for infinitely many primes, $p, 10p+1, 10p+3, 10p+7, 10p+9, 10p+A$ are all prime...

*...and that for most of those sets there are no additional primes between $10p+1$ and $10p+A$ other than the ones specified in the set, so that the infinitely recurring difference of $d$ is usually between consecutive $x_i$.
