Totally Tetradic!

What are Tetradic Primes?

Tetradic primes are a special type of palindromic prime which have the additional property of being both left-right symmetric and top-bottom symmetric. A consequence of this is that they have rotational symmetry too. Obviously, they only consist of the digits 0, 1, and 8.

The smallest tetradic prime is 11, which is unique in having an even number of digits [why] (in a similar way to 2 being the only even prime). Therefore the first tetradic prime that isn't this special case is 101, hence the logo.

Why are you searching for them?

I'm not. You are, hopefully!

Not being blessed with an infinite quantity of CPU power, I simply don't have the resources to search for them, but decided that they deserved a little bit more attention than they currently receive. Or should I say that Raffi Chaglassian decided this, and I agreed with him.

So, what's their significance?

They look cool. There's nothing mathematically special about them &emdash; it's simply that in their usual base-10 representation they're pretty little blighters. Or big blighters, as there's nothing preventing the discovery of enormous examples.

What's the plan?

To search for primes of the form 10^m+k*10^n+1 which are tetradic. Obviously, m and n are related (m is roughly twice n), and k is itself tetradic. For example:

10^48+18881*10^22+1 = 1000000000000000000000188810000000000000000000001
10^58+18081*10^27+1 = 10000000000000000000000000018081000000000000000000000000001

There are two methods of approaching such a search: One is to fix m (and n), and vary k. This is the method espoused by Harvey Dubner, who, until this project started, was the record holder. The other method, the one that this project presently uses, involves fixing k and varying m (and thus n). The fun with this latter method is that not all ks are equal! Some produce absolutely no primes at all (for example k=1 generates only numbers divisible by 3), and others generate particularly dense families.

Using a quick little sieve that I wrote specifically for the task, on literally millions of k values (so while each one is a quick sieve, the whole task is momentous) Raffi Chaglassian has found many k values which generate unusually dense families. And that's where you come in! Read on.

Can I have a block?

You certainly may. Pick a k value from the list below; contact me to let me know you'll be testing it; sieve the range, presumably using my 'kpal' sieve; and then test using PFGW. Then let me know when you've done. Of course, submit your primes to Professor Caldwell's top5000 primes list too!

The range that I recommend searching is n=30000-50000. However, I think that at least some of the ranges should be searched to n=70000. Any prime found with n>~60000 will become the world's largest palindromic prime, not just the world's largest tetradic.

Notes on density and yield

Note - the density in the table above is the ratio of candidates that remain in the range n=30000-50000 (so producing potentially 60000- to 100000-digit primes) that remain after sieving to 666666. Assuming responsible sieving, up to 440G (=666666^2), you'd only expect half this number of candidates to remain. So, a density of 0.3443 will probably leave you with 3443 candidates to test in the 30000-50000 range.

Amusingly, the density in the table doesn't tell you anything about how likely each candidate that survives sieving is to be prime. Sieving levels the field completely. What it does tell you is how many candidates are likely to remain after sieving, and therefore how large a task PRP testing will be, and the expected prime yield. Quite simply, the more candidates that remain after sieving, the more primes you're likely to find in that range, but the longer it will take to test, as there are more candidates to test.

It appears that the density of primes in candidates sieved to 440G ranges from 1/2900 at n=30000, to 1/4800 at n=50000. If you do the maths, you'll find that this means that a range with density 0.3770 should contain on average exactly one prime. Therefore all of the above ranges are expected, on average, to yield just more than one prime.

So get out there and find them!

Greetz

• Harvey Dubner, the godfather of tetradic primes.
• Raffi Chaglassian, for kicking this project into action.
• Anna, for the bloody stupid tetris idea. And stuff.
• Me, for being an arrogant and most cool Maths Bastard. (Anna wrote that.)
• Tristan ja Isolda, the cafe tucked into the corner of Raekoda in Tallinn, Estonia, which provided a delightful venue in which to write this page.

Create a handful of even-length tetradic numbers, and factor them. Can you see a pattern?