Amateurs work

these primes are called emirps:

en.wikipedia.org/wiki/Emirp

thanks guys, I hadn't caught up with your blogs for a while, and I'd been wondering what irrelevant opinions you'd come up with lately

You ask whether there is a prime p such that when p is "turned around" is also prime, and show that 13 is such a prime. Doing so you answered your own question.

worthless comment

> waaah why isn't anyone taking my thread seriously

No, I think it was quite an informative comment. Does it exist a prime number between 3 and 13? I know 5 is such a prime.

the fuck you mean "turned around"
why the hell would anyone be interested in this?
do you have autism

Shouldn't you be optimizing some pipes or something?

your mom optimized my pipe
very decent liquid flow

I hear your liquid flow is so optimized you can get the whole act done in less than a minute

only when its your mom

Well that's the only thing you can get on xbox live, so it doesnt really prove anything.

This is not an engineering thread.
This is a math thread about a very specific thing. If you hold no interest in said thing, don't shitpost here and please leave.

implying OP wasnt shitposting when he made this thread

He wasn't, this is a base-dependent property of primes, which fits my description of "very specific math thing". This is Veeky Forums, so math discussion is not shitposting.

Yes there are many such. You can try making your own computer program to find them. It is a good exercise in programming that will keep your brain sharp.

This.
Even the most trivial things are already named and studied.

Don't forget 11

>guaranteed replies

Another prime in base ten which (trivially) satisfies the OP's property, by dint of the fact that it is already a palindromic number, is

[math] \displaystyle 1000000000000066600000000000001 [/math]

This is known as /Belphegor's prime/. Tangentially related to some digits that OP posted earlier, notice that this integer is itself a 31-digit prime in base ten, while either side of number contains 13 consecutive 0-digits. Since this palindromic prime contains 666 in the middle, it has been cutely dubbed an instance of a /beastly palindromic prime/. Another example of a beastly palindromic prime, then, is

[math] \displaystyle 700666007 [/math]

But I don't know whether the above is the /smallest/ example of a beastly palindromic prime, which leads me to my idea for programmers: write a program that identifies the first few beastly palindromic primes.

While we are on the subject of "666", the number's obvious cultural importance is attended by some legitimately amusing number-theoretic properties. 666 is the 36 or 37th triangular number, depending on how you want to look at it (I had forgotten this), and is also the sum of the squares of the first seven primes, viz

[math] \displaystyle

\sum\limits_{p \; prime}^{7} p^2 = 666

[/math]

Furthermore, 666 is the sum (again, all in base ten, of course) of the first /gross/, that is, the first 144 /decimal digits/ of π. Chucking out the integer part of 3, this means: 1 + 4 + 1 + 5 + 9 + ... D_(144) = 666. There is a succinct, non-iterative way of expressing this which involves incrementing powers of ten and floor functions, I will probably amuse myself with finding the expression for it again.

Still, all of this chicanery assumes that we never leave base ten, as another user has rightly (aesthetically) pointed out. A simple result that immediately comes to mind is the /basis representation theorem/, front-and-center, page 8 in George Andrews' Number Theory, which can be got at any decent bookstore.

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

Amusingly, the smallest candidate for a beastly palindromic prime that I could think of OTTOMH is

[math] 16661 [/math]

Which WA informs me is in fact prime. Guess I answered my own question. (76667 also checks out).

(Andrews') Basis Representation Theorem simply reassures us that every natural number has a /unique/ basis representation, where /any other natural number/ within reason (>1) may be used as a base if we so choose. Having established that this is the case, Andrews then leaves it alone for much of the rest of the text, I guess. He conveniently side-steps the pedantic "but what about tally-marks!" issue for base 1, by simply not treating of base 1. Since he is writing a mathematics text, he also does not wax philosophical about arbitrarily many distinct symbols, as opposed to simply indexing a or k.

11 is a palindromic prime and not an emirp
mathworld.wolfram.com/Emirp.html

As for my above claim about pi's first 666 digits, there is first an algorithm to be descibed, and the moment it's been done, the finite series is apparent.

1) let k = 1. Then...
2) Start with π. 3.14159...
3) Multiply by 10^k = 10^1 = 10. 31.4159...
4) Subtract 30, which is precisely 10 times the floor of 10^(k-1) π, or 10 times the floor of π in this case.
5) Take the floor; it is 1 in this case. This is the digit out to such-and-such a place. Add this digit to the total.
6) While k < 144, now increment k. (or k++ as I suppose the programmers say).
7) Goto 2, or somesuch literal pseudocode, in my case.

(just once more, so we're getting it)

2') Start with π. 3.14159...
3') Multiply by 10^2 = 100. 314.159...
4') Subtract 310, which is 10 times the floor of 10(^1) π in this case.
5') Take the floor, it is 4. Add to total, (we have 5 so far).
6') While k

Since I've spent a little time on this, here is a reasonably good, non-crank page with more amusements on same. As before, we stay in base 10.

cadaeic.net/666.htm

Yes this is exactly what I was thinking about.

This is literally a project Euler question.