# What is Trinity Hall Prime number?

It is a prime number with 1350 many digits. I did not get much information about this number on the internet.

Question : What is Trinity Hall Prime number?

I watched this video but did not get the number.

It is a prime number with lots of $1$s and $8$s and some $0$s and other digits which looks like the coat of arms of Trinity Hall, Cambridge (not to be confused with its bigger neighbour Trinity College, Cambridge) and which has $1350$ digits the date of founding of Trinity Hall

• Whoever came up with this had far too much time on their hands! Sep 8, 2017 at 7:55
• Too bad the 621 is buried in there too.
– Paul
Sep 8, 2017 at 13:19
• @Hanno - Thank you. It was a screenshot from the video around 2:12, but you seem to be correct, and the framed version around 0:12 seems to have the 1350 digit version. I think there are even more differences than those you pointed out, and I have edited the picture Sep 8, 2017 at 17:18
• $\ddot\smile$ much appreciated your quick amendment; it's more substantial to have it right here at math.SE than on utube! Going to delete my previous comment as is has lost its purpose. Sep 8, 2017 at 17:46
• @TripeHound From the description under the Numberphile Video: "Professor McKee [the one responsible for the number] 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.'"
– Ovi
Sep 8, 2017 at 22:43

I took the time to copy down 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


If you want it as one number:

888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888111111111111111111111111888888111111111111111111111111888888111111811111111118111111888888111118811111111118811111888888111188811111111118881111888888111188811111111118881111888888111888811111111118888111888888111888881111111188888111888888111888888111111888888111888888111888888888888888888111888888111888888888888888888111888888111888888888888888888111888888811188888888888888881118888188811188888888888888881118881188881118888888888888811188881118888111888888888888111888811111888811118888888811118888111111188881111111111111188881111111118888111111111111888811111111111888811111111118888111111111111188881111111188881111111111111118888811118888811111111111111111888881188888111111111111111111118888888811111111111111111111111888888111111111111111111111111118811111111111111111111111111111111111111111111062100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001


The video seems to explain what the prime number is. There is also a a bit more in the description.

The bigger question is probably how Professor McKee came up with this number. According to the description to the video, McKee fixed most of the digits so that the emblem was displayed. It sounds like he then chose some digits and just tried a number of different digits until he got a prime.

I think the following answer: https://mathoverflow.net/questions/27508/factors-of-p-1-when-p-is-prime might address why the $p-1$ was chosen with many factors.

• "+1" for as a one number
– user275490
Sep 7, 2017 at 17:56
• Hmmmm. I expected McKee would've taken the start of the zeros sequence and tried, sequentially, 1000..., 2000..., etc, and tested for primality, up until 0621 gave a prime, but 1680 also gives a prime and it occurs earlier.
– E.P.
Sep 8, 2017 at 14:41

Basically, it's like ASCII art, but with a still more limited "palette," of the coat of arms of Trinity Hall, Cambridge, though with the added bonus that interpreted on a single line as one number it's a prime number.

You too can make yourself that kind of ASCII art for your own coat of arms or other such symbol. First, compose the digits in the pattern that you want. Then, in a program like Mathematica, use the NextPrime function. Provided you don't make the number too large, you should be able to get the answer in a few seconds.

Heck, even Wolfram Alpha doesn't take too long to respond to NextPrime[10^1350] with $10^{1350} + 271$. But you might run into problems trying to send Wolfram Alpha the number from your ASCII art. So you need Mathematica on your own computer.

You might need to tweak the last few digits, i.e., if you made the last line short by mistake.

I saw the actual thing in real life in the Senior Combination Room at Trinity Hall (Cambridge, England):

When junior research fellows get to the end of their stint (usually after three years), they traditionally give some gift to the college that represents "something about them". In this case, that "something" is an ASCII number in the shape of the coat of arms of the college. The number 1350 (the number of digits) is significant as it's the year that the college was founded by Bishop Bateman.

It is in a wooden frame, displayed among other pictures, paintings, and the trappings of a Cambridge college.

Here is another one that Jose Zaman made for Code-Golf site of stackexchange PPCG
https://codegolf.stackexchange.com/q/146017/67961

it looks nice and doesn't have the "annoying fix" in it

777777777777777777777777777777777777777
777777777777777777777777777777777777777
777777777777777777777777777777777777777
777777777777777777777777777777777777777
111111111111111111111111111111111111111
111111111111111111111111111111111111111
188888888118888888811188888811188888811
188111118818811111881881111881881111881
188111118818811111881881111111881111111
188888888118888888811881111111881118888
188111111118811111111881111111881111881
188111111118811111111881111881881111881
188111111118811111111188888811188888811
111111111111111111111111111111111111111
111111111111111111111111111111111111111
333333333333333333333333333333333333333

• This doesn't answer the question Mar 30, 2019 at 21:32
• Do you mean that ALL the other answers answer the question? I think you should read all the other answer carefully. They just add info to the first answer. And this is exactly what this answer does by giving an other example. Other examples is a fundamental tool in order to explain something. In this case you can see that this prime is not a unique achievment but you can make your own pretty easilly. As a result the reader is motivated to search deeper into huge primes and see how rare they are. Mar 30, 2019 at 22:23
• Oh, sorry for the trouble. I am also on PPCG Mar 30, 2019 at 22:49