Sunday Dec 21, 2025
Episode 11 - Cyclic Orifice
In which we discuss orifices, pure maths, and the world's best 'spy'.
If you want to view the Interesting Scale (where we rank our facts), click here! If the link doesn't work, then copy this address: https://www.legendkeeper.com/app/cmgq90c870bcj0jlh0gsi6ml3/nzd6hyhw/
Music in the episode: "Flying Kerfuffle" and "Space Jazz" Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 4.0 License
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3 months ago
I meant that p divides 10^(p-1) -1. So 7 divides 10^6-1 = 999999. (There doesn’t appear to be a way to edit a comment)
3 months ago
You’ll notice that all the numbers I’ve given have that property of the second half being the ”complement” of the first half.
3 months ago
This works because a prime p divides 10^(p-1). (That’s Fermat’s Little Theorem). 1/7 and 1/17 are fully cyclic decimals, but 1/13 is only 6 digits long rather than 12, so there are two different patterns in the cycle. This is because 13 happens to divide 999999 as well as 999999999999.
3 months ago
And that is why 142857 is the only non -trivial one. You get this pattern for the reciprocal of any prime (eg 7, 13, and 17). But the only primes less than 10 are 2, 3, 5, and 7. 2 and 5 are factors or 10, so have terminating decimals. 3 has a trivial cycle of length 1. And anything above 10 will have a leading zero. So 1/7 is the only example that works.
3 months ago
1/17, on the other hand, is .0588235294117647 recurring, and that does work with consecutive multiples. However, both of these have leading zeros, so won’t work as cyclic integers.
3 months ago
What I didn’t know was that this is the only non-trivial cyclic number, and I thought about why. 1/13 is .076923 recurring, and multiples of it are either cycles of it or a second pattern .153846 recurring, so it’s one is the ones Michael referred to as non -consecutive multiples.
3 months ago
I immediately recognized 142857 as the recurring part of the decimal expansion of 1/7, and knew about the properties Michael went on to describe. One he didn’t mention is that if you take the first three digits, 142, and subtract each from 9, you get the other three, 857.