There's this book I borrowed a while ago called the Dictionary of Curious and Interesting Numbers by David Wells, and even though I've long since given it back, I have continued the practice of writing about certain numbers and their curious properties. Today, it's the turn of 280, a number which has a Wikipedia page consisting of only four lines.
280
Whilst 280 may seem unremarkable on the surface due to its meagre four lines, it's part of the exclusive club of idoneal numbers - there are only 65, and 280 is the 52nd largest. These numbers came about after Euler wanted to find very large primes, and the ones that did the job were ideonal (these numbers are also called suitable or convenient numbers). The obvious question is what an idoneal number is - and I hope this is a simple enough explanation:
- An idoneal number is a positive integer D where x2 ± Dy2, where x2 and Dy2 are relatively prime to each other (they share only 1 as a common factor), is either a prime power or twice a prime power. A prime power is a prime number raised to a certain power. For x=3, y=1, and D = 280, you get 32 + 280*12, which is 289, or 172 - so a prime power.
- There is another, far nicer definition for idoneal number - if a number cannot be represented in the form of ab + bc + ca, where a, b, and c are all distinct positive integers, it's ideonal. So for instance 11 isn't ideonal as if you plug in 1, 2 and 3, you get 1*3 + 1*2 + 2*3 = 11,
The main question from here, however, is how many ideonal numbers are there. We know of 65, with the largest being 1848; Euler proved 1848 was the largest below 10,000, and it surprised him. Due to a proof by Weinberger, we know there are at most two additional ideonal numbers - and so we're stuck in an unsatisfying quandary.
ζ(3)
This number has a value of 1.20205..., and is irrational. It's derived from this infinite series:
This is also known as the zeta function, and it can get rather interesting – here, s is a given integer, and the output would thus be designated as ζ(s):
ζ(0) and ζ(1) aren’t anything special – ζ(0) is just 1+1+1+1+…, which is divergent. Same goes with ζ(1), which is just 1 + ½ + 1/3 + ¼ +…, so therefore there’s no numerical sum of either of them.
Things get interesting, though, with ζ(2). As s gets smaller, one would expect the series to converge, and indeed it does here. Mathematicians were stumped as to what this equalled in what was known as the Basel problem, and it remained that way until Euler solved it. The solution…π2/6. It’s unsurprisingly also irrational, as π is irrational.
Then you get to ζ(3), which is another unique constant – this one is called Apery’s constant, after the mathematician Roger Apery. Using triple integrals, he managed to prove that it was, too, irrational. To this day, one still doesn’t know if ζ(3) is transcendental, however, so that’s something you could do this weekend.
From here on in, ζ(s) will converge to 1, as the values in the expansion get increasingly smaller, and as such the various values of ζ(s) get less and less notable.
The solution for sin x = x is rather obvious, x = 0. It's worth noting we're dealing with radians here, as well - in degrees, x will vary (not in the case of sin x = x, though).
When cos x = x, though? Things get a bit more tricky. See, I could work out that sin x = x at x = 0 using an infinite Maclaurin expansion:
- sin x = x - x3/3! + x5/5! - ...
Factorising out an x:
- sin x = x(1 - x2/3! + x4/5! - ... )
For sin x = x, we need 1 - x2/3! + x4/5! - ... = 1, and that only happens if x = 0, hence the previous answer.
cos x is a bit more irritating; as I mentioned in a previous blogpost:
- cos x = 1 - x2/2! + x4/4! - ...
We can't do the x factorising trick, and even if we did it would get rather messy. So it shouldn't be shocking that the solution for cos x = x was found by a professor spamming their calculator who noticed that when they input a given number into the cos function, and they always ended up at the same number - or 0.739085...
If you attempted to solve by using the Maclaurin expansion, a cheap trick would be to discard all terms with powers greater than 2. This will enable you to get a quadratic equation: x2 +2x -2 = 0. Through this, we get a good approximation for the Dottie number, where x = -1 + √3, which is 0.732050... - compared to the actual value, you get a percentage difference of 0.95%!
If you want a symbol for it, a paper by a researcher named H. Arakelian denoted the Armenian letter ա. This is one of few instances where I've seen a non-Latin or Greek letter used to represent a constant.
Favourite number survey
A few years ago, I did a survey where I asked people in my school what their favourite number was. After various taunting by people, I compiled a list of 188 numbers - some are clearly satirical responses (how did someone think 1004081 was a good response, it's not even prime) but I did find out that 31 people, or about a sixth, chose 7 as their favourite. Which is to be expected - 7 commonly appears as a candidate for favourite number. It's possibly because 7 comes about a lot in religion, culture, and is known as a lucky number, and it's the first polysyllabic number. Or people said it to shut me up. 3 and 4, at twelve submissions each, were the only numbers to appear more than ten times. The data is insignificant, but I've been sitting on it for three years, so I decided to discuss it here now.
And that's it for another series of number facts. Here's to 350, which doesn't even have a Wikipedia page...in fact, up to 1000, the only multiples of 70 that have Wikipedia pages from here are 420 and 700, and I reckon the former is due to a different reason. Not that finding number facts will be difficult or anything...
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