I can come up with more examples and so could Pythagoras
I forgot how to search with my phone. I didn’t see 163 in there?
Please show me your best example of those numbers together. That does not relate to a triangle.
From another Dope board:
Write down any three digits. Write the same digits again in the same order next to the originals. Take this six-digit number and divide by 143. The result will always be a whole number - no remainder.
This only works for you. But you seem to be acting as if it is evidence of some sort of universal law.
I mean, here’s the thing: your thread is about ‘coincidences that should only occur rarely’. This one does. You’re one of 7.5 billion people on Earth, and as far as we know, you’re the only person for which this is true. I’d say that’s occurring rarely.
Sorry — let’s back up — you do want Pythagorean triples, possibly satisfying some additional property?, or you want some other kind of triples, or…?
There is a formula for J(τ) there; if you substitute τ = (1 + √-163)/2 you (very not coincidentally) get an integer.
Let’s pretend I’m in 2nd grade. Where does 3,4 and 5 show up together in a formula, in math, or some constant? In some result of counting. The answer to a formula when you plug in real numbers that happen. The result of division or multiplication. You don’t have to remind me that it shows up when you are counting 1,2,3,4,5. That it involves triangles. Show me some case that I cannot explain. Even if it’s a random number generator in the first 4 digits. Do you understand what I’m looking for? Something magic. From God. I guess this does not exist. Maybe that’s the point. How often do patterns show up in the beginning page of e digits?
Even more amazing when my tax return adds up to an exact dollar amount, given that I do those only once a year! (And yes, it happened to me.)
ETA: And e is very nearly a rational number with a repeating decimal expansion! 2.718281828…
You’re on to something Mangetout. My simple formula may be the shortest or most elegant for any non-trivial 7 digit phone number. Literally 1 in a billion people. Or 10 million phone numbers. That makes Rick very special. Certainly it’s the shortest that is symmetric. You can only tie it, no matter how hard you try. Starting with a single phone number. Your friends number. But I’m not 1 in a Billion special. This implies something.
That it happens more than it should?
That we notice it more than we “should”?
If I’m the only person with a cool short symmetric formula in the USA. What are the odds that I noticed it?
The astonishing part is that (assuming that even integer is greater than 4), you can make both of those odd integers primes!
(Probably. But if you can find one where it doesn’t work that way, you’ll be famous.)
Apparently, not really.
14/(9 - ∛121) = 3.45345…, is that enough of a random coincidence?
That’s good close to what I mean. But you made up that formula choosing constants to get the results I’m asking for. Let’s find a formula or constant or count that already exists?
Reminds me of one night in Sunnyvale when I realized the car in front of me was Feynman’s - It’s been a while but I believe the plate was TUVA.
It took me all of 3 seconds to see why that must always be true.
A number with the digits abcabc is equal to abc * 1001, and 1001 is a multiple of 143.
Here’s a similar observation: A “palindromic” number is one whose digits read the same in both directions, like 1374731. Some prime numbers are palindromic. I once compiled a lengthy list of them, and I noticed unusually large gaps: There were a bunch of 3-digit palindromic primes, and a bunch of 5-digit palindromic primes, but not a single 4-digit palindromic prime. I realized that there can not be any palindromic prime with an even number of digits other than 11, and proved it. I think I was in 11th or 12th grade when I did that.
Here’s another example where the odds of it happening are 26^3. My first girlfriend’s Dad, only ham friend since kindergarten, and a relative all have the exact same ham radio call sign after the 3. There is no way to explain this. It just is.
This is different than an IP address today, matching 1 of 100 relatives birthdays that I’ve known. I’ve known in real life very few ham operators.
I have discovered a wonderfully simple proof of this, which unfortunately I cannot fit into the margins of this post.
About 27,000 squared? Like winning a lottery?
I’ve got a magic trick involving math that will blow your minds. It’s math so it’s factual right? And it’s a question. So is this the right board to post it on? Will start a new thread.
So let’s look at this blended idea instead.
… How often do patterns show up in the beginning page of pi digits?
As soon as the OP can make a coherent statement about what a “pattern” is (and more importantly, isn’t) then and only then we can begin to get someplace.
A pattern is a mostly a psychological artifact. As such it’s as real as the creatures we see in clouds.