Trinity Hall Prime (from Numberphile)

[Mini-Geek]

I watched the above video and read McKee's explanation and wondered if his prime is the best Trinity Hall Prime.

Professor McKee explains: "Most of the digits of p were fixed so that: (i) the top two thirds made the desired pattern; (ii) the bottom third ensured that p-1 had a nice large (composite) factor F with the factorisation of F known. Numbers of this shape can easily be checked for primality. A small number of digits (you can see which!) were looped over until p was found that was prime."

I'll define "Trinity Hall Prime" as a 1350 digit prime number with the same 8's and 1's in the first 900 digits, and being provable by p-1 (having a trivial factorization for at least 33.3%) and leave the "best" part up to interpretation.

I couldn't find the prime online anywhere, so I transcribed the prime:

888888888888888888888888888888
888888888888888888888888888888
888888888888888888888888888888
888111111111111111111111111888
888111111111111111111111111888
888111111811111111118111111888
888111118811111111118811111888
888111188811111111118881111888
888111188811111111118881111888
888111888811111111118888111888
888111888881111111188888111888
888111888888111111888888111888
888111888888888888888888111888
888111888888888888888888111888
888111888888888888888888111888
888811188888888888888881118888
188811188888888888888881118881
188881118888888888888811188881
118888111888888888888111888811
111888811118888888811118888111
111188881111111111111188881111
111118888111111111111888811111
111111888811111111118888111111
111111188881111111188881111111
111111118888811118888811111111
111111111888881188888111111111
111111111118888888811111111111
111111111111888888111111111111
111111111111118811111111111111
111111111111111111111111111111
062100000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000001

Where C is the 900-digit constant that is the first 30 lines of the above, this can be written as C*10^450 + 621*10^446 + 1. As for how he arrived at this particular number? 621 is the first number you find if you search for such a number that might make a prime with 446 as the exponent. 446 is a reasonable value, as it is very likely that you'd find such a prime in your 10,000 options there, and not very likely you'd find it if you searched k*10^447 for k < 1000.

The existing prime has its 31st line start with "0621". I wondered if I could make it something like "xxx0" instead, so that in some sense, I am only messing with three digits instead of 4.

I conducted a search and found a match! C*10^450 + 168*10^447 + 1. It is a prime, provable the same way, but the 31st line is "168000...". I'd consider this a better prime than Prof. McKee's! There are no such primes with only 1 or 2 digits. There may be a Trinity Hall Prime with only 1 or 2 modified digits *elsewhere* in the block of 0's, but I haven't searched that yet, and it'd require a lot more luck for p-1 to still be trivially factored. Even at the start, you need a *bit* of luck to get the full 33.33...%, as the trivial power of 10 necessarily accounts for less than one third of the 1350 digits.

I found four numbers when I searched k*10^446 exhaustively (unless I goofed somewhere): 621, 1680, 2307, 3309.

[science_man_88]

https://math.stackexchange.com/quest...l-prime-number also came up after the video.

[ATH]

I found this prime, which changes the "imperfection" 621 to 4 symmetrical 1's on each side. It looks nicer imo. I'm running Primo on it now, but running 200 sprp tests with GMP says it is PRP, so the risk it is not prime is practically zero:

888888888888888888888888888888
888888888888888888888888888888
888888888888888888888888888888
888111111111111111111111111888
888111111111111111111111111888
888111111811111111118111111888
888111118811111111118811111888
888111188811111111118881111888
888111188811111111118881111888
888111888811111111118888111888
888111888881111111188888111888
888111888888111111888888111888
888111888888888888888888111888
888111888888888888888888111888
888111888888888888888888111888
888811188888888888888881118888
188811188888888888888881118881
188881118888888888888811188881
118888111888888888888111888811
111888811118888888811118888111
111188881111111111111188881111
111118888111111111111888811111
111111888811111111118888111111
111111188881111111188881111111
111111118888811118888811111111
111111111888881188888111111111
111111111118888888811111111111
111111111111888888111111111111
111111111111118811111111111111
111111111111111111111111111111
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
100000000000000000000000000001
000000000000000000000000000000
100000000000000000000000000001
000000000000000000000000000000
100000000000000000000000000001
000000000000000000000000000000
100000000000000000000000000001
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000001

[ATH]

Primo says "Candidate certified prime" as expected.

It only took 35 minutes on a single core, so there is no need to find numbers that can be proved with p-1 factorization. This was probably needed back then many years ago when Professor McKee found it.