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

Question

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)?

Answer 1

There is a big heuristic justification for why it is probably true. First though, instead of thinking about a primorial prime also being a palindrome, we can simply think of it as a primorial number also being a palindrome and the math still will show that your conjecture is probably true.

So, primorial numbers are of the form $p_n\# \pm 1$, but that form is not easy to deal with as primes are complicated, so we will approximate it with: $$p_n\# \pm 1 \sim e^n$$ So then let's think about the chance that a random number is a palindrome. Well, the definition of a palindrome is that the first half of the digits are the same as the reverse of the last half of the digits. Therefore if a number is about $10^x$ then the odds that it is a palindrome is $10^{x/2}$ because the first half of the digits can be anything, but the last half of the digits have to be match it. The equation $10^x \to \frac{1}{10^{x/2}}$ is equivalent to $x \to \frac{1}{\sqrt{x}}$ so well be using the $2$nd as it is easier to use.

Now let's consider asking how many primorial numbers are also a palindrome with the number $>10^{11}$, because there are no counterexamples to your conjecture at least up to over $10^{11}$. Well, starting with $n=11$ we get $p_n\# > 10^{11}$, so well write a sum starting with it: $$(\text{Approximate number of palindrome primorials} > 10^{11}) \approx \sum_{k=11}^\infty \frac{1}{\sqrt{e^x}}$$ That last step might be a little confusing. Basically, we are adding up the chances that the numbers are counterexamples to see how many (or little) we would expect.

Next well write $\text{(Approximate number of palindrome primorials > 10000)}$ as $C$ for count. Well also write $\sqrt{e^x}$ as $(\sqrt{e})^x$ to get:

$$C \approx \sum_{k=11}^\infty \frac{1}{(\sqrt{e})^x}$$

By solving the series we get: (Wolfram page here)

$$C \approx \frac{1}{(\sqrt{e} - 1)e^5}$$

Which means $C \approx 0.0103\dots$ which is less than a 1.1% chance!

To summarize what this is saying, is that the chance that besides the small examples you gave there is approximately less than a 1.1% chance that there is another. This could be dramatically improved by just checking a few more primorials, and remember, this heuristic did not even take into account that the numbers have to be prime!

If you have any questions, just ask!

Answer 2

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!

Answer 3

============ First Point ======================

I've noticed something rather unique to this specific palindrome pair :

• 16(7*)61

  and

• 151

It seems that, regardless of number of 7s, it's always divisible by 151

Conversely, for palindromes in the form of 1(6+)1

whenever it's a 2-factor only composite, and one of them being 11, then the other prime factor must be a palindromic prime in the form 1(51)+

I haven't tested all combinations of it, but this n versus n-1 relationship seems to be applicable

if n in [2:9]

Another similar type of repeating pattern, while not being palindromic, is that numbers in the form of 3(9*)6 are always(?) divisible by 36,

in addition,

 whenever the the `# of 9s is odd`, 

which is same as

 whenever `# of decimal digits is odd` for this set,

then it's also divisible by 11 ; underlying principle being

% echo '       11*36' | bc 
              396
% echo '      111*36' | bc
             3996
% echo '     1111*36' | bc 
            39996
% echo '    11111*36' | bc 
           399996
% echo '   111111*36' | bc 
          3999996
% echo '  1111111*36' | bc 
         39999996
% echo ' 11111111*36' | bc 
        399999996

================== Unrelated Second Point ============

Perhaps other examples also exist, for now I've only noticed that divisibility by palindromic primes of 3 or 11 are what I call

horizontally-digit-agnostic (HDA)

meaning -

• horizontally reversing the decimal

  expansion of the digits of an integer,

  does not, alter its divisibility by n

e.g.

3614599901: 11 2423 135617
1099954163: 7 11 1733 8243

modulo % 3 seems to be also HDA, but % 11 isn't

here's a summary of other small primes evaluated against the HDA criteria

• 2 fails it by 4x : 16 ->61, 61 is prime

• 5/7 fails it by 5x/7x : 35 ->53, 53 is prime (also 20->02,30->03)

• 13/31 fails this on itself : 13<->31, 13/31 both primes

• 17/71 fails this on itself : 17<->71, 17/71 both primes

• 37/73 fails this on itself : 37<->73, 37/73 both primes

• 19 fails it by 2x : 38 ->83, 83 is prime

• 23 fails it by 4x : 92 ->29, 29 is prime