210 and Other Numbers

A while ago, I borrowed a book called the Dictionary of Curious and Interesting Numbers by David Wells, and I even got two blogposts out of it. However, I've long since given that book back, but I'm willing to keep the tradition going and to signpost my 210th post with another look at various numerical information. As is tradition, I'll start off with:

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 (δ_S) 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.