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.

111 0.1119
8 0.1942Fabio D. M.testing...
10111100100111101 0.3443Raffitesting34234, 39463
1180108011880881108010811 0.4087
8180880100888880010880818 0.4093
11100181011011018100111 0.4096
1011818080808080808181101 0.4117
81088810011811001888018 0.4131
1180818888118118888180811 0.4133
1008088010018100108808001 0.4143
88808800111111100880888 0.4179
1800818001810181008180081 0.4183
1110088088811188808800111 0.4191
8188181188000008811818818 0.4198
1181180001100011000811811 0.4205
1080000881001001880000801 0.4215
8180180101011101010810818 0.4216
18188181088088018188181 0.4230
1081101080188810801011801 0.4334Jacktesting

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!


[* why is 11 the only tetradic prime having an even number of digits?]

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

Another hastily constructed page by Phil Carmody
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