# The Corpus Christi Prime

This number is a prime, with 2688 digits. It also looks rather a lot like Corpus Christi College, Cambridge. The top left corner encodes my initials, JRH, in ASCII. The bottom right corner is my date of birth.

I was inspired by Numberphile’s most recent video, which demonstrates a prime number of 1350 digits, which looks like the coat of arms of Trinity Hall College, Cambridge.

I have some nice postcards with this number on it. If you live in Cambridge and would like some, send me an email.

I created some pixel art, which looks a bit like the College, but slightly squished since pixels are square, but characters are rectangles.

Next, I selected a font (Menlo) and counted the number of pixels used for each digit. I chose Menlo because it has very heavy zeros, which I thought might come in useful.

Then I wrote a program which generated an “ideal” number, based on these two pieces of information. I then manually made the two required modifications (which were not completely narcissistic – the number had to end with an odd digit, and numbers starting with 1 are ever so slightly more likely to be prime).

Finally, I generated random fluctuations in the number and tested each with the Miller-Rabin primality test. This produced a shortlist of numbers which were very very likely to be prime. I used Dario Alpern’s fantastic tool to determine whether any of them actually were prime. Of the 8 candidates I had generated overnight, all of them were prime, so I selected the nicest looking one, which you see above.

### Why Was I Confident That This Would Work?

The prime number theorem tells us that there are approximately $\frac{n}{\log (n)}$ primes less than $n$. So there are approximately

$\frac{10^{2688}}{\log 10^{2688}} - \frac{10^{2687}}{\log 10^{2687}} \approx 1.6 \times 10^{2683}$

2688-digit primes. So approximately one in every 6200 2688-digit numbers is prime. Now, I wasn’t looking at even numbers, so that reduces that number by half, so things are looking quite good.

I set my program running for a little bit and determined that on my hardware (a MacBook Air with a 1.7 GHz processor and 8GB of RAM) my program could determine whether about 30 2688-digit numbers are probably prime per minute. So I thought that it would take about 100 minutes to find a candidate. I had slightly overestimated this time: overnight (9 hours), I checked 25750 numbers and found 8 probable primes. This lines up pretty well with the back of the envelope calculation above.

### The Number

Feel free to check whether it is prime:

100101077777777777777077777777777777777777077777777777777777777710100107777777777777020777777777777777777020777777777777777777771001000777777777777770777777777777777777770777777777777777777777777777777777777777770507777777777777777770507777777777777777777777777777777777777770551077777777777777770155077777777777777777777777777777777777777055507777777777777777055507777777777777777777777777777777777777700003777777700777777730000777777777777777777777777777777777777704333077777708807777770333407777777777777777777777777778777777777000007777771001777777000007777777777777777777777877707777777777709990777771055017777709990777777777770777777777777770177777777770999077771055550177770999077777777771077777777777777047777777777099907717053003507177100007777777777407777777777777010777777778700080050051111115005008000777777777701077777777777701077777777771099000511000000115000999077777777770107777777777777247777777777000001115051111605111001007777777777327777777777777700777778777709930150011500511005103990777777777700777777777777770778777777710993000111108801111000399017777877777077777777777777047777777700000001111115005111112000000078777777407778777700070056011001105505990111111111111111109950550110011065007000706560655550055005550090011111160051111110090055500550055556065605555555555555555555009001115088988805111009005555555555555555555000000550000000000009090111908888880911109090000000000005500000011111055011111111110909011108888888801110909011111111110550111119511105501115995111090901118008008008111090901115995111055011159085110550116800851103030111888888888811103030115900851206501158008011000011080080110535011188888888891110535011080080110000110800801105501108008010900001118888888888111000090108018011055011080080210550110800801105550111000033000011105650110800801105501108008011055011080080110595034000000000999330595011080080110550110803331105501133333311093305555655555555555033901133333312055011333000000550000000000009590999999999999999909590000000000005500000011115055051111111110949011111121111111110949011111111150550511111111105501111111111092901119000100009111092901111111111055011112111110000111111211109590121088898888011109590111111111100001111185111055011158911110959011108008800801111959011119851110550111580051106501150008111092901110088888800111092901118000511055011500083110550113808011005590111018888880011109550011080831105501138008311055011380801105555501100888888001105555501108083110560113900031105611130000110555550110088888800110555550110000311055011300111105555011111111055555011008888880011055555011111111055550111211110555501111111105555501100888888001105555501111111105500000000000055550000000000555550000088888800000555550000000000550120397

## 14 thoughts on “The Corpus Christi Prime”

1. Well done.

So you are only 19 or 20, depending on whether your date of birth is written in crazy US format or at-least-in-order-of-magnitude most-of-rest-of-world format?!

PS: My website went into a coma soon after it was born.

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This is fascinating stuff. Prime numbers have always interested me.
By chance I discovered a way to factorize a number without having to divide anything. It turns out some easily-generated sequences have such a relationship that unless a term from one can be found in the other, then the value of n used to generate one of those sequences must be prime. These sequences need only a number of terms equal to a quarter of n and, thanks to a handy mathematical law it, each term need not have any more than a very low number of digits no matter how large the value of n. It seems to me it would be very easy for a powerful computer to generate and sort through the two sets of data looking for a match. Much easier than the current method of trying to divide huge numbers into huge numbers. Do you have any thoughts on this?

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1. Thanks! Your test reminds me of the Lucas primality test, and “thanks to a handy mathematical law it, each term need not have any more than a very low number of digits no matter how large the value of n” is reminiscent of modular arithmetic. If you’d like me to have a look in more detail, feel free to email me: my crsid (in the sidebar) @cam.ac.uk

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The sequences that are generated involve no modular arithmetic. I discovered the procedure by using a very simple (non factor-related) algorithm to represent integers and discovered the pattern always corresponded to their known factors and prime numbers always looked ‘disordered’ compared to composites. Certainly modular arithmetic is used to more easily and quickly generate this graphic representation, but the algorithm works on its own. From this discovery I eventually derived the aforementioned 2-sequence procedure which I have proved infallibly identifies primes or can identify prime factors of n using knowledge of which matching terms produce ‘hits’.
I’m not a mathematician and the mathematical reasoning behind this procedure is beyond my current ability to express in academic language; however, I will try to find a way of making it as clear as possible and get back to you. Thank you so much for your interest!

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3. Grey Baker says:

This is rad. Nice work.

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1. Tom Brunt says:

It’s not entirely dissimilar to the front facade of QEGS…

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4. This is really astounding. It inspired me to write a program of my own to fully automate this process, from which I’ve generated a few more primes- including a 8946 digit prime in the shape of my school’s chapel.

It’s all open source and can be seen at
https://www.github.com/tim-clifford/pixel-prime

I’d be intersted to see how your code compares to mine =]

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5. Grahame says:

Hi Jack,
I’m an old Corpuscle (matriculated in 1986), and was wondering if I could have an image of the whole Corpus Christi Prime to print and put on the wall of my office, please?
Good luck with your studies, and enjoy your time at Corpus.
Grahame

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1. Dear Graham,

I’m afraid that this image is the largest one I have. However, if you email me your address, I’ll send you a postcard that you can put up!

Best,

Jack

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