Speaking of which, every so often someone discovers a new prime number (often a Mersenne prime).
We’ve known for a very long time that there are infinitely many prime numbers (there is no Largest Prime Ever), but at any given time, there’s a largest known prime number, and it’s big news when a larger one than that is discovered.
For a time, there was the little known number of wisserteen. But the number was eliminated in 2012 as being too divisive.
Now I’m tempted to request an example of something that will never be posted on this message board.
Well, one could be clever about it, and say something like “the number you get by adding 1 to 100130211006 has no google hits”. But I’m sure some joker here is going to ruin it by replying with the number in their post.
–Mark
There’s a page with something to do with yoga in the URL that has a long list of integers. But it doesn’t have 92615000000 + 2. And I’m not bored enough to find any smaller.
Indeed, since real numbers were algebraically extended to complex numbers, which in turn were generalized into the quaternions, and even those broadened into the octonions, I keep wondering if/when someone will devise a logically consistent system of hexadecions. (Unless we know that’s not possible).
It’s not possible. The octonions are as far as it goes.
But you lose properties the further up you go. The octonions aren’t associative, but are alternative. The sedenions aren’t even alternative.
There are remarkable numbers still turning up. Numberphile makes a big deal out of Mill’s constant, A = 1.3063778838630806904686144926… with the property that ⌊ A[sup]3[sup]n[/sup][/sup]⌋ is prime for all natural numbers n.
However, there are infinitely many real numbers A with that property. I think the prime-generation capability of Euler’s lucky number 41 is more impressive (n[sup]2[/sup] + n + 41 is prime for all natural n < 40). Particularly impressive is the way Euler’s lucky number leads to Martin Gardner’s April Fool’s Integer
e[sup]π√(4*41-1)[/sup] = 262537412640768743.999999999999…
Playing around just now, I squared the April Fool’s Integer (M) to get M[sup]2[/sup] = 68925893036109279891085639286943768.0000000001637…
At first I expected a number with fractional part .999… but of course the reciprocal of M is much smaller than its difference from an integer so that didn’t make sense. But why does M[sup]2[/sup] have fractional part .0000000001… ? The odds of a random real being so close to an integer are less than one in a billion. Coincidence? I doubt it: I checked another Heegner number ( e[sup]π√(4*17-1)[/sup]) and see a similar result.
Is this a well-known result? I shan’t play on Indistinguishable’s patience since I’d be at a loss just to explain the closeness of the April Fool’s result itself. :o
(pdf) Here’s one mathematician’s list of the ten “coolest” numbers. He has the Heegner number 163 in the #1 slot. I was intrigued by his #8, a Sierpinski number: Upthread was a formula that generates only prime numbers when n is a natural number, but (using a Sierpinski number) the formula
78557*2[sup]n[/sup] + 1generates only composites!
I think that’s a flaw in whatever tool you used to do the calculation.
262537412640768743.999999999999250 squared is
68925893036109279891085639286943729.8810388468840000000000…
Your answer differs after 33 digits.
–Mark
The number I’ve underlined above is not the exact number. (I wrote “…”). If you have a tool good enough to multiply such precise numbers, it can probably produce a preciser approximation to
e[sup]π√(4*41-1)[/sup] = 262537412640768743.9999999999992500725971981856888793538563373369908627…
BTW, not only is the square of this number very close to an integer, but so are all of its small powers, as well as its cube root.