210
It's the product of the first four prime numbers, for one, and can thus also be designated by 4#. It's a primorial, which is much like a factorial except instead of multiplying a consecutive sequence of natural numbers from 1, you multiply a consecutive sequence of primes from 2. The first few primorials are:
- 1# = 2
- 2# = 2*3 = 6
- 3# = 2*3*5 = 30
- 4# = 2*3*5*7 = 210
- 5# = 2*3*5*7*11 = 2310
and so on. I'd assume 0# is 2#/2, or 1, for much the same reason that 0! is 1!/1, and Wikipedia seems to agree with me on this.
Much like how factorials can be represented by the gamma function [Г(x)], primorials are also linked to the first Chebyshev function, [Ө(x)], where Ө(x) = ln(p#).
Collatz conjecture
The Collatz conjecture goes like this:
- Think of a number
- If it's even, divide by 2
- If it's odd, multiply by 3 and add 1
- All numbers should end up, after a series of iterations, at 1.
Every integer up to 2.95x1020 has been proven to satisfy these conditions, mainly through using computers - yet maths is all to rigorous, and thus the Collatz conjecture remains unproven.
9 has a particularly lengthy sequence - it has the longest out of the first ten numbers - which goes as follows:
9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
To have a lengthy sequence, you should hope to stray far from a power of 2, which also means you need to stay clear of numbers like 5 and 21, since you'll end up on one after the next iteration. As a result, it's rather easy to see how you'll end up at 1 in the end. I've played around with each of the numbers from 1 to 9 before, which would be the final digit of a number, and all of them ended up at 1 in the end, but I doubt the proof was rigorous enough.
√2 and the silver ratio
The most famous irrational number is maybe √2, because it's arguably the first such number. Easily constructed from a square with length 1, before drawing the hypotenuse, it's said Hippasus proved it was irrational by contradiction, before dying gruesomely as a consequence. Yet the silver ratio takes √2 a bit further...it adds 1 on to it.
The silver ratio is 2.4142135..., and can also be defined by the repeating fraction 2+1/(2+1/(2+1/...), or [2,2,2,2,...]. Much like the golden ratio, the silver ratio has its own unique spiral, is given its own Greek letter (δS) and has its own geometric definition, where (2a+b)/a = a/b, where a is a side length longer than b.
Therefore, 2 + b/a = a/b, so if we let b/a equal x:
- 2 + x = 1/x (multiplying through by x)
- 2x + x2 = 1
- x2 + 2x - 1 = 0
- x is 1∓√2
It lacks the mystic nature of φ, since no musicians or architects or sunflowers have quoted the silver ratio, yet it's curious enough. However, all these ratios merely play to a general formula, where for a number N, if (Na+b)/a = a/b, the ratio equals (N + √N2 + 4)/2 - in this case, the silver ratio is for N = 2. I found this website, which has a proof for this general form.
There are other ratios, though. ς represents the supersilver ratio, since it's the solution to x3 = 2x2 + 1, which resembles the silver ratio equation from before, and ρ is the plastic ratio, the solution to x3 = x + 1, neither related to that general formula.
Gnomonic numbers
These are all the odd numbers, owing to the fact that gnomonic numbers are derived from the area of a gnomon. A gnomon is the red shape, and is formed by removing a square from a larger square:
Unsurprisingly, we end up with a difference of two squares, which is of the form a2 - b2, or (a+b)(a-b).
- Odd numbers can be represented as 2n+1
- This is of the form n2 + 2n + 1 - n2, which is (n+1)2 - n2, thus every odd number can be written as the difference of two squares.
- By definition, all the gnomonic numbers are the odd numbers.
At least, that's according to this website - the OEIS lists many other sequences of gnomonic numbers.
Sphenic numbers
Since I started with primes, I may as well end with them. This category has come up a lot in my research, and it's defined as a number which is the product of three distinct primes. The smallest is also 3#, or 30, and from there, you get others like 70 (2*5*7) and 2431 (11*13*17) - there being an infinite number of primes, there will also be an infinite number of sphenic numbers. Why they're called sphenic (wedge-shaped) seems to be a mystery.
Semiprimes are similar, except they're the product of two (distinct) primes - the smallest is 22, or 4, yet if you like distinction, it would be 2*3 = 6. With semiprimes, you can draw a rectangle of any area in only one distinct way, and as such they come up in puzzles (where no, you don't get exactly 1000 puzzle pieces in a box) and cryptography, since they have two easy to identify factors. They also came up in the Arecibo Message, a radio transmission broadcast into space in the faint hope a distant civilisation uncovers us. The message was a 23*73 rectangle, and supposedly those civilisations would quickly realise that we used a semiprime, as primes are universal. Whether they'd get the rest of the message is questionable.
I reckon that's enough numbers for now - perhaps I'll come back with some more before 280 rolls around.
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