# What relationship(s) [if any] exist between primorial primes and palindromic primes?

Information on primorial primes are in the following hyperlinks:

MathWorld - Primorial Prime

Wikipedia - Primorial Prime

On the other hand, we have the following hyperlinks providing information on palindromic primes:

MathWorld - Palindromic Prime

Wikipedia - Palindromic Prime

My question at this point would be: What relationship(s) [if any] exist between primorial primes and palindromic primes?

The first few primorial primes are

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209

The first few palindromic primes in base-10 are:

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181 … sequence A002385 in OEIS

From these two lists, the following "conjecture" appears plausible:

$\mathbf{CONJECTURE}$: In base-10, the only prime numbers that are both primorial primes and palindromic primes are 3, 5 and 7.

Is this conjecture known in the literature? If so, does it have a name (i.e., has it been proposed before)?

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It is fair to question, what is your heuristic justification behind this conjecture? –  al-Hwarizmi Aug 16 '13 at 16:48

Let me try it.

(1) You state: "the first few primorial primes are..." from here we conclude that there exist more which not taken into account here (incomplete).

(2) You state: "The first few palindromic primes in base-10 are..." from here we conclude that there exist more which not taken into account here (incomplete).

(3) You then state (without any serious heuristic justification) based on a glance on the incomplete sequences listed in (1) and (2) that it appears plausible to state such being a conjecture; and that only what our eye obsevers up to some number in (1) and (2) would suffice to be appearing plausible.

Such statement, from a mathematician point of view shall not be termed as conjecture as long as no serious heuristic justification provided.

To your questions follows the answer based on the 3 steps above:

What relationship(s) [if any] exist between primorial primes and palindromic primes? No there is no explicit relationship, at least not any that is even heuristically justifiable.

Is this conjecture known in the literature? No there is not existing such statement. And honestly it would be rather absurd to believe that some matematician seriousely would make such statement termed as conjecture in the future except with a proper heuristic foundation.

If so, does it have a name? No indeed not.

Let me take another view on the standpoint for such a statement. Imagine, even if this would have been a conjecture based on a strong heuristic justification, and we could also provide a proof later on. So you would have then a theorem that would say $3,5,7$ are the only primes both palindromic and primorial. This would be as little spectacular as saying today that it would be a significant theorem to state that: the only integers which are both palindromic (or primorial) and Fermat number are $3,5$. Would that make sense? No!

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Thank you for your (somewhat) detailed answer @al-Hwarizmi. At any rate, my heuristic justification is that a quick visual examination of both lists give you (somewhat immediately) $3$, $5$ and $7$. Notwithstanding, I suggest that you ask for the motivation for this question, rather than the heuristic justification. =) –  Arnie Dris Oct 10 '13 at 17:25