Conway’s Subprime Fibs

Siobhan Roberts’ biography “Genius At Play; The Curious Mind Of John Horton Conway” is about 400 pages long, and for a while I thought I’d never finish it.   In fact, I rather thought I might never properly start it. You’ll guess, correctly, that some of the content is far beyond my grasp, but a far more important reason is that I kept putting the book on one side while I did actually play with some of the ideas. Most notably, I was stuck for weeks at the very start, page 2 in fact.

Conway’s “Subprime Fibs” was wholly new to me. It’s a concoction with elements of the Fibonacci sequence, prime numbers, and the Collatz Conjecture, and it’s wholly accessible to pupils in the early years of secondary school. Based on my experience, you’ll need lots of squared paper, a pencil, and a good eraser – it’s humblingly easy to make a slip. A table of prime numbers is helpful as well, though you’ll find your ability to recognise a prime number improves rapidly.

Here’s the presentation from Roberts’ book:

Take 2 numbers, and write them down. Then add them up. If the sum is a prime number – a number divisible only by 1 or itself – then leave the number alone and write the number down. If the sum is not prime then divide it by its smallest prime divisor and write down the resulting number. And then take the last 2 numbers written down add them up, repeat the process, and carry on.

1 and 1 make 2, and that’s prime [1 1 2]

1 and 2 make 3, and that’s prime [1 1 2 3]

2 and 3 make 5, which is prime [1 1 2 3 5]

3 and 5 make 8, which isn’t prime, so I divide it by the smallest prime I can, which is 2, and I get 4 [1 1 2 3 5 4]

5 and 4 make 9, which isn’t prime; divide it by 3 and I’ll get 3, which is prime [1 1 2 3 5 4 3]

4 and 3 make 7, which is prime [1 1 2 3 5 4 3 7]

3 and 7 make 10, which I can divide by 2 [1 1 2 3 5 4 3 7 5]

7 and 5 make 12, which I can divide by 2 [1 1 2 3 5 4 3 7 5 6]

5 and 6 make 11, which is prime [1 1 2 3 5 4 3 7 5 6 11]

….

Here’s the 1,1,…. sequence continued. Like a Collatz sequence it rises and falls unpredictably:

1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, 15, 23, 19, 21, 20, 41, 61, 51, 56, 107, 163, 135, 149, 142, 97, 239, 168, 37, 41, 39, 40, 79, 17, 48, 13, 61, 37, 49, 43, 46, 89, 45, 67, 56, 41, 97, 69, 83, 76, 53, 43, 48, 13, …

The same number may occur more than once, with the sequence continuing in different ways (for example, 61) but its memory goes back only two terms, so as soon as you get two successive terms repeated its structure is fixed from then onwards. So at 48, 13, …. it enters a repeating loop of 17 terms.

This feels pretty Collatzy, doesn’t it, and prompts questions whether all pairs of number result in a loop, whether different loops are possible, how long it takes for loops to appear, ….

Not surprisingly, Subprime Fibs have been explored, but as yet nowhere near as extensively as the Collatz Conjecture. It’s been suggested that every sequence eventually cycles, but I don’t think this has yet been proved.

And if you want to try totally unexplored territory, it looks tempting to try some kind of Tribonacci sequence, where each term is the sum of the three previous terms. So a Tribonacci sequence would be 1,1,1,3,5,9,17,31,57,….

My first attempt at Tribonacci Fibs gives 1, 1, 1, 3, 5, 3, 11, 19, 11, 41, 71, 41, 51, 163, 51, 53, 89, 193, 67, 349, 203, 619, 1171, 1993, 1261, 1475, 4729, 1493, 179, 173, 615, 967, 585, 197, 583, 455, 247, 257, 137, 641, 345, 1123, 703, 167, 1993, 409, 367, 923, 1699, 427, 3049, 1725, 743, 1839, 73, 885, 2797, 751, 403, 1317, 353, 691, 787, 1831, 1103, 61, 599, 43, 37, 97, 59, 193, 349, ….

I’m not sure how interesting this is. I can think several different reasons to suggest that perhaps we’re not actually going to get any interesting cycles, but you’re welcome to prove me wrong.

There’s a footnote which I ought to offer. Playing with these ideas Conway-inspired ideas has ensured I’m considerably less likely to feel comfortably smug about my mental arithmetic.   I don’t feel too bad about failing to recognise a multiple of 19, but on occasion I’ve certainly failed to notice a multiple of 11, or, shamefully, a multiple of 3. I’ve also made misadditions of two or three numbers.   Each slip gave me considerable sympathy with William Shanks, who spent twenty years calculating pi to 707 decimal places. He published in 1873, but I’m glad he never knew he’d gone wrong at position 528.

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