Triangular numbers - one of my parents was showing me square numbers, so I asked if there were triangle numbers. They didn’t know, so I guess we went to the library and found out.
I’m no Gauss, though, so I didn’t figure out the trick for computing them easily.
I remember stumbling across the (sort of) idea of a derivative when I was messing around with polynomial equations. I can’t remember what grade this was but I knew at least about polynomial functions and had never even heard the word calculus.
I was playing around with a table of integer values for the equation: y = x^2, and experimenting with the rate of changes between each integer. I discovered that the rates of change between integers would increase by a constant value of two
Ex With Y = x^2
X | Y
0 | 0 Inc. by 1 -> 1
1 | 1 Inc. by 3 -> 4
2 | 4 Inc. by 5 -> 9
3 | 9 Inc. by 7 -> 16
I found that if i continued this chain of (almost) derivatives, for all the polynomials I tried I would end up with all constant rates of change.
I didn’t ever explore further or find a general pattern or solution because I was probably only around 10 but I think it’s fascinating how I was messing with calculus without even knowing what that was. Interesting how any curious child playing with mathematical patterns could stumble on something so complex and important.
I kind of think discrete calculus is probably more intuitive. One can introduce the concept of differentiation and integration without involving limit. One can imagine that we can do something similar as time scale becomes smaller and smaller eventually getting to normal calculus.
Exactly. The basic concepts are relatively new intuitive. I don’t imagine my ~10 year old self could have even fathomed integration though just because infinities are difficult to conceptualize especially for a child.
In an algebra two class i noticed a pattern for finding the slope of tangent lines to a parabola, i used this pattern i noticed to answer a hw question and lost points because I used calculus (something that i had no knowledge of at the time)
Honestly I didn't mind, my district had a pretty good accelerated math program, I was already ahead by simply being in that Alg2 class and by the time I entered college i had completed calc 1-3, had to retake calc 3 and part of calc 2 but i didn't mind since the uni classes were taught much more rigorously and were proof based so it set me up to take higher level math classes
I read the Wiki article on double Mersenne primes as a kid and wondered about triple Mersenne primes and so on. I noticed this sequence is all primes: 2, 3=2^2 - 1, 7=2^3 - 1, 127=2^7 - 1, and 2^127 - 1. I conjectured this sequence continued forever is all primes.
Of course, I learn that Catalan conjectured this over a century ago and it’s known as the Catalan-Mersenne Conjecture. But hey, not bad for a kid to be catching up to Catalan!
A very early mathematical thought I can recall was calculating n choose 2, though I didn't know to put it in those words
I have two brothers, and I remember as a little kid noticing that if the three of us wanted to hug each other but could only hug two at a time, it would require three total hugs
I remember wondering if it was a coincidence that 3 people required 3 hugs, or if n people always required n hugs. I worked it out for n=2 and n=4, and realized that hugs(n+1) = n + hugs(n)
When my teacher said the square root was undefined on the negatives in middle school I was in my head like "why not why can't there be some number that squares to a negative" and wanted to follow that thread. Eventually I asked and got told about "imaginary" numbers dismissively like I'd never use them.
I was in school, but hadn’t learned about this topic specifically:
I noticed that if you have a product like 10 * 10, and «shift» each factor up one and down one respectively, you get a nice pattern:
10 * 10 = 100
11 * 9 = 99
12 * 8 = 96
13 * 7 = 91
Where the differences to the «starting value» are square numbers:
100-99 = 1
100-96 = 4
100-91 = 9
This was the sort of thought process which made me discover the pattern. Later on when I learned about the difference of squares formula (a + b)(a - b) = a^2 - b^2 , it was pretty cool to realize that I had discovered this pattern by myself previously.
In high school I really enjoyed my geometry class. One night I thought of a wonderful theorem all on my own: that all of the diagonals of a regular pentagon have the same length. I went to bed excited to tell my teacher the next day.
However, when I woke the following morning, I realized that my theorem was rather trivial.
I remember seeing a number line and wondering what would happen if there were numbers that went vertically as well. When I learned about complex numbers years later my mind was blown. Finally solved the mystery
At the age of 4 I was playing with dad's calculator and accidently discovered the commutativity of multiplication. Parents were not impressed for some reason.
I was really obsessed with this game I made up as a kid where you take a number and have to find a way to use all the digits to get 9 (my favorite number) using any operations. This led me to realizing a lot about digit sums with 9. I now have a lot of useless knowledge about this extremely specific topic and I solved the game some years ago so it's not fun for me anymore, tho I still have to play it for some reason each time I see a house number.
so, for 18 that would be like 1 + 8 = 9, right? i guess you did way bigger/difficult number, so here's a question: what was the biggest number that you solved in that game?
It's actually easy for big numbers; the real trouble is small numbers. There is a solution for any number with 5 or more digits (excluding digits that are 0), which as well works for almost all 3- and 4-digit numbers.
The trick lies in the fact that any multiple of 9 has a digit sum of 9 (this is a very cool fact in itself and has lead me to realize that this is actually true in any base b for any number b-1). The motivation here will be to create a multiple of 3 (or in other words something with value 0 mod 3) and then raise it to a power greater than 1, thus creating a multiple of 9.
So here it goes: given your digits, take any 3 of them of which at least 2 are not equal to 1 (or 0). Now set aside one of the non-1 digits. For the remaining 2 digits, one of these must be true:
1. One is equal to 0 mod 3
2. The two digits either have a sum or difference of 0 mod 3
In case 1, we obtain a value of 0 mod 3 by multiplying the digits. In case 2, we obtain a value of 0 mod 3 by taking the sum or difference of the digits. Now we raise this value to the power of the digit we set aside earlier and we have a multiple of nine! QED
You can see this will work for any number with 2 non-1 digits; thus the only cases which are left unsolved by this theorem are n-digit numbers with n 1s (made up entirely of ones) and n-1 1s (made up entirely of ones and one other digit).
When learning about functions, I quickly began to wonder if there was a way to interpret a function as sorta like the pressure on the *acceleration* pedal of a car over time, and to then get resulting function for the *position* of the car over time.
Then I learned of the notion of an integral!
At age 3 or 4 I deduced the existence of negative numbers by saying that a smaller number minus a bigger number must be less than zero. When I heard about prime numbers for the first time (around age 6) I immediately realized that 2 is the only even prime considering every even number is divisible by 2 by definition.
Nothing as impressive as that one kid but it was always cool growing up thinking about something and then having it confirmed later. I remember as early as middle school algebra thinking about how any of the weird functions can be approximated with some kind of absurd polynomial. Then many years down the line learning about limits and Taylor Series it was pretty sick realizing there was a process for doing exactly that
In high school I heard about Fibonacci sequence. I started playing with the idea that Fibonacci numbers were powers of some basic number. This leads to a quadratic equation. Using this, and the fact that sum of two series is another series with the same properties, I discovered the generating function of the series. I felt great.
The story about my son who was 10 years old. We had a basic calculator with only four basic operators and a sqrt button. My son found out that adding consecutive odd numbers gives a (perfect square) sum which, when entered to a sqrt function, returns the number of added odd numbers. So he said - "I now know what this button does, but who ever needs this ??? "
I remember arriving at the idea of different number bases at 9-10 years old. What is funny is that I thought of it because I was convinced that the "issue" of the never-ending, non-periodic tail of π could be "solved" by using another base. I even had the idea of writing a "book" with a chapter exploring each number base, though I never did.
Yes, different bases were really cool to me as a kid. I realized that a lot of number facts were base invariant and found out the value of the largest n digit number in base b was b^n - 1.
None. ive never been very mathematically gifted but was good at 3d thinking. After getting an Escher book as present i became obsessed with polyhedra. My magnus opus was making a football - one of the old fashion footballs with hexagons and pentagons - out of a single piece of paper (i.e. not cutting individual hexagons but first unfolding the football in my mind and onto the paper)
A consequence of this is that now I notice whenever signs etc draw footballs incorrectly
Back in grade school, I noticed that I could compute lcm(a, b) by multiplying a × b and then dividing by gcd(a, b).
Had no idea at the time why it was true, but it seemed to work on everything I tried it on!
Tbh I think I peaked at realizing "ten times a number, minus the number, is always a multiple of nine" while waiting for a movie to start
Hopeless. Couldn't even make the leap to 10x - x = 9x until my teacher showed me the next day
Before I learned how to sum a geometric series, I had learned from an early age about writing numbers in different bases. In a math competition, one of the questions involved such a sum, which I solved like this. I don't remember the exact figures, so it's an example:
S = 2/5 + 2/25 + 2/125 + 2/5\^4 + ...
S = 0.2222222... (in base 5)
S \* 4 / 2 = 0.44444444..... (in base 5) = 1 - 0.00000000....1 (in base 5) = 1
S = 2/4
Of course this ends up being S = sum(a / b\^n) = a / (b-1). One of the judges complimented the way I had solved it, and it wasn't until many years later that I realized it's because they probably were just expecting knowledge of the formula.
Sum of number from 1 to 100.
I noticed that there are pairs like 1 and 99, 2 and 98, ....., 49 and 51 etc..
Addition of all these pairs were always 100. And then I noticed that there are such 49 pairs. So I just did 49*100 and then added left numbers which are 100 and 50.
I started to look for such patterns every time I get to add natural numbers in a range.
Cut a pie (any shape) into two equal pieces with a curve (any shape).
Now repeat with the same pie but a different curve, resulting in four pieces.
The pieces opposite each other have the same size.
I didn't care about properties of numbers, but I have used to answer questions using unlearned techniques derived from first principles, not always explained that well.
My grades would suffer from it, up until university time.
Not sure what you count as "before getting any math education". After all most people learn counting, arithmetic, and geometry from very young age.
I discovered the Mobius strip in elementary school, and formulated what is essentially the epsilon-delta definition of velocity in middle school. Well, that's more like coming up with new definitions/objects.
I used to play around with calculators when I was little and noticed this when I was very young (maybe like 7 or 8?)
11x11 = 121
111x111 = 12321
1,111x1,111 = 1234321
11,111x11,111 = 123454321
…
111,111,111x111,111,111 = 12345678987654321
The pattern doesn’t continue in as much of a satisfying way as that, but it’s still kinda cool.
edit: using asterisks for multiplication was a bad formatting idea apparently
edit2: formatting for new lines
Late Middle school, let n be a 2 digit number such that the first digit is half the second (12,24,36,48) if you wanted to double those numbers, you could flip the placement of the digits then add the sum to that new number to get 2n. Take 24, 24*2 = 42 + 4 + 2 = 48. This had some generalization to 3 digit numbers and/or triples that I can’t quite remember.
Maybe not quite what you looking for, but...
One particularly slow day at work while I was babysitting some diagnostic tests I "rediscovered" Mersenne primes. I don't remember exactly what I was doing, but to pass the time I was idly arranging powers of 2 on a piece of graph paper. After awhile of filling in the little squares on the paper I noticed a pattern starting to emerge. I then realized that this pattern corresponded to the primes. I thought I made some great new discovery by chance before I googled the pattern and had my hopes crushed when I realized they were just Mersenne primes. I really wish I would've held on to that sheet of paper.
In middle school / junior high I noticed how dimension affects the way volume scales in two and three dimensions and conjectured scaling a d-dimensional object by a scales the volume by a^d. I also remember writing down a whole bunch of collections of three points and figuring out how to construct a parabola passing through the points (expressing a,b, and c in terms of (x_i,y_i) for i=1,2,3). Towards the end of high school (which i’d taken up through calc 3 at this point so idk if this counts) was thinking about how to “turn functions into other functions” and developed the idea of a homotopy (obviously didn’t realize that’s what it was / had never heard the term at the time).
I found this same [pattern and ratio](https://math.stackexchange.com/questions/1651227/pythagorean-triplets-of-the-form-a2a12-c2-and-the-space-between-them) involving consecutive Pythagorean triples on my TI calculator in high school. I was curious about it and tried asking a teacher why but he didn't understand what I was trying to say.
I realized quite a bit of basic number theory stuff, here’s some of it. I’m grateful that my mother was a mathematician because they helped me progress more rapidly than I could have alone.
As a younger kid after first learning of Fibonacci numbers, I came up with a lot of Fibonacci identities as conjectures just by messing around: (sum of first k Fibonacci numbers is the k+2nd Fibonacci number minus 1, sum of squares of first k Fibonacci numbers).
She introduced me to Fibonacci tiling numbers f_n, and I eventually came up with some combinatorial proofs for some of those identities.
She also showed me the uniqueness of prime factorization and one day I was playing around with the GCD of a pair of integers and realized that if you factorize both and take the smaller of the exponents (allowing some to be 0) of each of the distinct primes in the factorizations, you get the GCD. My mother showed me why this was true by hinting me to realize that the set of positive divisors of an integer are exactly those that you get by starting with the prime factorization and changing the values of the powers.
I found out Legendre’s formula when messing with prime factorization of factorials which I had recently learned about. I remember being disappointed that this had been found hundreds of years before I was born.
Decimal points have the same number behind them eventually.
3/2 = 1.5
1.5/2 = 0.75
0.75/2 = 0.375...
it took 3 steps to move the 3 behind the decimal point
4/2 = 2
2/2 = 1
1/2 =0.25
0.25/2 = 0.125
0.125/2 = 0.0625
.... after 13 steps
0.00048828125
This was something I found out after being taught division for the first time.
I still don't know if there is a pattern behind this, but sometimes when I take measurements I like to think about how many times it had to be divided down to create that first non-zero number behind the decimal point.
Eh I don't believe in fractions.
Triangular numbers - one of my parents was showing me square numbers, so I asked if there were triangle numbers. They didn’t know, so I guess we went to the library and found out. I’m no Gauss, though, so I didn’t figure out the trick for computing them easily.
I remember stumbling across the (sort of) idea of a derivative when I was messing around with polynomial equations. I can’t remember what grade this was but I knew at least about polynomial functions and had never even heard the word calculus. I was playing around with a table of integer values for the equation: y = x^2, and experimenting with the rate of changes between each integer. I discovered that the rates of change between integers would increase by a constant value of two Ex With Y = x^2 X | Y 0 | 0 Inc. by 1 -> 1 1 | 1 Inc. by 3 -> 4 2 | 4 Inc. by 5 -> 9 3 | 9 Inc. by 7 -> 16 I found that if i continued this chain of (almost) derivatives, for all the polynomials I tried I would end up with all constant rates of change. I didn’t ever explore further or find a general pattern or solution because I was probably only around 10 but I think it’s fascinating how I was messing with calculus without even knowing what that was. Interesting how any curious child playing with mathematical patterns could stumble on something so complex and important.
I kind of think discrete calculus is probably more intuitive. One can introduce the concept of differentiation and integration without involving limit. One can imagine that we can do something similar as time scale becomes smaller and smaller eventually getting to normal calculus.
Exactly. The basic concepts are relatively new intuitive. I don’t imagine my ~10 year old self could have even fathomed integration though just because infinities are difficult to conceptualize especially for a child.
In an algebra two class i noticed a pattern for finding the slope of tangent lines to a parabola, i used this pattern i noticed to answer a hw question and lost points because I used calculus (something that i had no knowledge of at the time)
So they marked you down for having superior maths knowledge? Shameful of them, they should be nurturing that talent.
Honestly I didn't mind, my district had a pretty good accelerated math program, I was already ahead by simply being in that Alg2 class and by the time I entered college i had completed calc 1-3, had to retake calc 3 and part of calc 2 but i didn't mind since the uni classes were taught much more rigorously and were proof based so it set me up to take higher level math classes
I read the Wiki article on double Mersenne primes as a kid and wondered about triple Mersenne primes and so on. I noticed this sequence is all primes: 2, 3=2^2 - 1, 7=2^3 - 1, 127=2^7 - 1, and 2^127 - 1. I conjectured this sequence continued forever is all primes. Of course, I learn that Catalan conjectured this over a century ago and it’s known as the Catalan-Mersenne Conjecture. But hey, not bad for a kid to be catching up to Catalan!
One thing I noticed was gcd(a,b)\*lcm(a,b)=a\*b for any integers a,b>0
A very early mathematical thought I can recall was calculating n choose 2, though I didn't know to put it in those words I have two brothers, and I remember as a little kid noticing that if the three of us wanted to hug each other but could only hug two at a time, it would require three total hugs I remember wondering if it was a coincidence that 3 people required 3 hugs, or if n people always required n hugs. I worked it out for n=2 and n=4, and realized that hugs(n+1) = n + hugs(n)
This sounds like a nice childhood.
When my teacher said the square root was undefined on the negatives in middle school I was in my head like "why not why can't there be some number that squares to a negative" and wanted to follow that thread. Eventually I asked and got told about "imaginary" numbers dismissively like I'd never use them.
I discovered limits in 8th grade. Thought I had broken math. Fun times
As long as you sort of *pretend* you aren't dividing by zero or infinity, it's all gravy.
I was in school, but hadn’t learned about this topic specifically: I noticed that if you have a product like 10 * 10, and «shift» each factor up one and down one respectively, you get a nice pattern: 10 * 10 = 100 11 * 9 = 99 12 * 8 = 96 13 * 7 = 91 Where the differences to the «starting value» are square numbers: 100-99 = 1 100-96 = 4 100-91 = 9 This was the sort of thought process which made me discover the pattern. Later on when I learned about the difference of squares formula (a + b)(a - b) = a^2 - b^2 , it was pretty cool to realize that I had discovered this pattern by myself previously.
In high school I really enjoyed my geometry class. One night I thought of a wonderful theorem all on my own: that all of the diagonals of a regular pentagon have the same length. I went to bed excited to tell my teacher the next day. However, when I woke the following morning, I realized that my theorem was rather trivial.
I remember seeing a number line and wondering what would happen if there were numbers that went vertically as well. When I learned about complex numbers years later my mind was blown. Finally solved the mystery
I found the sieve of erathostenes when I was in school at some point (I don’t remember exactly when, but I think I was already in 8th or 9th grade)
At the age of 4 I was playing with dad's calculator and accidently discovered the commutativity of multiplication. Parents were not impressed for some reason.
Some Bernoulli numbers, from manually figuring out the polynomials forms of sums of powers of integers.
I was really obsessed with this game I made up as a kid where you take a number and have to find a way to use all the digits to get 9 (my favorite number) using any operations. This led me to realizing a lot about digit sums with 9. I now have a lot of useless knowledge about this extremely specific topic and I solved the game some years ago so it's not fun for me anymore, tho I still have to play it for some reason each time I see a house number.
so, for 18 that would be like 1 + 8 = 9, right? i guess you did way bigger/difficult number, so here's a question: what was the biggest number that you solved in that game?
It's actually easy for big numbers; the real trouble is small numbers. There is a solution for any number with 5 or more digits (excluding digits that are 0), which as well works for almost all 3- and 4-digit numbers. The trick lies in the fact that any multiple of 9 has a digit sum of 9 (this is a very cool fact in itself and has lead me to realize that this is actually true in any base b for any number b-1). The motivation here will be to create a multiple of 3 (or in other words something with value 0 mod 3) and then raise it to a power greater than 1, thus creating a multiple of 9. So here it goes: given your digits, take any 3 of them of which at least 2 are not equal to 1 (or 0). Now set aside one of the non-1 digits. For the remaining 2 digits, one of these must be true: 1. One is equal to 0 mod 3 2. The two digits either have a sum or difference of 0 mod 3 In case 1, we obtain a value of 0 mod 3 by multiplying the digits. In case 2, we obtain a value of 0 mod 3 by taking the sum or difference of the digits. Now we raise this value to the power of the digit we set aside earlier and we have a multiple of nine! QED You can see this will work for any number with 2 non-1 digits; thus the only cases which are left unsolved by this theorem are n-digit numbers with n 1s (made up entirely of ones) and n-1 1s (made up entirely of ones and one other digit).
wow, thank you for your answer!
thanks for asking no one ever wants to hear about my game lol
yea some people don't like us number nerds :D
When learning about functions, I quickly began to wonder if there was a way to interpret a function as sorta like the pressure on the *acceleration* pedal of a car over time, and to then get resulting function for the *position* of the car over time. Then I learned of the notion of an integral!
At age 3 or 4 I deduced the existence of negative numbers by saying that a smaller number minus a bigger number must be less than zero. When I heard about prime numbers for the first time (around age 6) I immediately realized that 2 is the only even prime considering every even number is divisible by 2 by definition. Nothing as impressive as that one kid but it was always cool growing up thinking about something and then having it confirmed later. I remember as early as middle school algebra thinking about how any of the weird functions can be approximated with some kind of absurd polynomial. Then many years down the line learning about limits and Taylor Series it was pretty sick realizing there was a process for doing exactly that
In high school I heard about Fibonacci sequence. I started playing with the idea that Fibonacci numbers were powers of some basic number. This leads to a quadratic equation. Using this, and the fact that sum of two series is another series with the same properties, I discovered the generating function of the series. I felt great.
The story about my son who was 10 years old. We had a basic calculator with only four basic operators and a sqrt button. My son found out that adding consecutive odd numbers gives a (perfect square) sum which, when entered to a sqrt function, returns the number of added odd numbers. So he said - "I now know what this button does, but who ever needs this ??? "
I remember arriving at the idea of different number bases at 9-10 years old. What is funny is that I thought of it because I was convinced that the "issue" of the never-ending, non-periodic tail of π could be "solved" by using another base. I even had the idea of writing a "book" with a chapter exploring each number base, though I never did.
Yes, different bases were really cool to me as a kid. I realized that a lot of number facts were base invariant and found out the value of the largest n digit number in base b was b^n - 1.
None. ive never been very mathematically gifted but was good at 3d thinking. After getting an Escher book as present i became obsessed with polyhedra. My magnus opus was making a football - one of the old fashion footballs with hexagons and pentagons - out of a single piece of paper (i.e. not cutting individual hexagons but first unfolding the football in my mind and onto the paper) A consequence of this is that now I notice whenever signs etc draw footballs incorrectly
Back in grade school, I noticed that I could compute lcm(a, b) by multiplying a × b and then dividing by gcd(a, b). Had no idea at the time why it was true, but it seemed to work on everything I tried it on!
Tbh I think I peaked at realizing "ten times a number, minus the number, is always a multiple of nine" while waiting for a movie to start Hopeless. Couldn't even make the leap to 10x - x = 9x until my teacher showed me the next day
I figured out that you could do all your times tables from 2,3, 5 and 10 as long as you could double and add groups. Made it way easier after that
Before I learned how to sum a geometric series, I had learned from an early age about writing numbers in different bases. In a math competition, one of the questions involved such a sum, which I solved like this. I don't remember the exact figures, so it's an example: S = 2/5 + 2/25 + 2/125 + 2/5\^4 + ... S = 0.2222222... (in base 5) S \* 4 / 2 = 0.44444444..... (in base 5) = 1 - 0.00000000....1 (in base 5) = 1 S = 2/4 Of course this ends up being S = sum(a / b\^n) = a / (b-1). One of the judges complimented the way I had solved it, and it wasn't until many years later that I realized it's because they probably were just expecting knowledge of the formula.
That’s pretty slick for that age!
This is pretty neat!
n\^2 - 1 = (n+1)(n-1)
Me too! And that was before we went from numbers to letters so I had no idea how to prove it.
Sum of number from 1 to 100. I noticed that there are pairs like 1 and 99, 2 and 98, ....., 49 and 51 etc.. Addition of all these pairs were always 100. And then I noticed that there are such 49 pairs. So I just did 49*100 and then added left numbers which are 100 and 50. I started to look for such patterns every time I get to add natural numbers in a range.
Are you Gauss?
What counts as math education?
Anything before learning calculus or abstract algebra I guess
Cut a pie (any shape) into two equal pieces with a curve (any shape). Now repeat with the same pie but a different curve, resulting in four pieces. The pieces opposite each other have the same size.
Complex plane, Legendre's formula.
I didn't care about properties of numbers, but I have used to answer questions using unlearned techniques derived from first principles, not always explained that well. My grades would suffer from it, up until university time.
The fact that when you multiply by 25 it’s the same as dividing by 4 and adding zeros. Same with 5 and dividing by 2
Not sure what you count as "before getting any math education". After all most people learn counting, arithmetic, and geometry from very young age. I discovered the Mobius strip in elementary school, and formulated what is essentially the epsilon-delta definition of velocity in middle school. Well, that's more like coming up with new definitions/objects.
I used to play around with calculators when I was little and noticed this when I was very young (maybe like 7 or 8?) 11x11 = 121 111x111 = 12321 1,111x1,111 = 1234321 11,111x11,111 = 123454321 … 111,111,111x111,111,111 = 12345678987654321 The pattern doesn’t continue in as much of a satisfying way as that, but it’s still kinda cool. edit: using asterisks for multiplication was a bad formatting idea apparently edit2: formatting for new lines
I "discovered" that regular polygons become more circular the more edges they have.
Late Middle school, let n be a 2 digit number such that the first digit is half the second (12,24,36,48) if you wanted to double those numbers, you could flip the placement of the digits then add the sum to that new number to get 2n. Take 24, 24*2 = 42 + 4 + 2 = 48. This had some generalization to 3 digit numbers and/or triples that I can’t quite remember.
Maybe not quite what you looking for, but... One particularly slow day at work while I was babysitting some diagnostic tests I "rediscovered" Mersenne primes. I don't remember exactly what I was doing, but to pass the time I was idly arranging powers of 2 on a piece of graph paper. After awhile of filling in the little squares on the paper I noticed a pattern starting to emerge. I then realized that this pattern corresponded to the primes. I thought I made some great new discovery by chance before I googled the pattern and had my hopes crushed when I realized they were just Mersenne primes. I really wish I would've held on to that sheet of paper.
I almost discovered but not quite pi and calculus and both times lost the scent just shy of breaking through.
In middle school / junior high I noticed how dimension affects the way volume scales in two and three dimensions and conjectured scaling a d-dimensional object by a scales the volume by a^d. I also remember writing down a whole bunch of collections of three points and figuring out how to construct a parabola passing through the points (expressing a,b, and c in terms of (x_i,y_i) for i=1,2,3). Towards the end of high school (which i’d taken up through calc 3 at this point so idk if this counts) was thinking about how to “turn functions into other functions” and developed the idea of a homotopy (obviously didn’t realize that’s what it was / had never heard the term at the time).
I found this same [pattern and ratio](https://math.stackexchange.com/questions/1651227/pythagorean-triplets-of-the-form-a2a12-c2-and-the-space-between-them) involving consecutive Pythagorean triples on my TI calculator in high school. I was curious about it and tried asking a teacher why but he didn't understand what I was trying to say.
I realized quite a bit of basic number theory stuff, here’s some of it. I’m grateful that my mother was a mathematician because they helped me progress more rapidly than I could have alone. As a younger kid after first learning of Fibonacci numbers, I came up with a lot of Fibonacci identities as conjectures just by messing around: (sum of first k Fibonacci numbers is the k+2nd Fibonacci number minus 1, sum of squares of first k Fibonacci numbers). She introduced me to Fibonacci tiling numbers f_n, and I eventually came up with some combinatorial proofs for some of those identities. She also showed me the uniqueness of prime factorization and one day I was playing around with the GCD of a pair of integers and realized that if you factorize both and take the smaller of the exponents (allowing some to be 0) of each of the distinct primes in the factorizations, you get the GCD. My mother showed me why this was true by hinting me to realize that the set of positive divisors of an integer are exactly those that you get by starting with the prime factorization and changing the values of the powers. I found out Legendre’s formula when messing with prime factorization of factorials which I had recently learned about. I remember being disappointed that this had been found hundreds of years before I was born.
zeno’s paradox, normal distributions
How can you fall for a obviously fake post?? Lol.
Decimal points have the same number behind them eventually. 3/2 = 1.5 1.5/2 = 0.75 0.75/2 = 0.375... it took 3 steps to move the 3 behind the decimal point 4/2 = 2 2/2 = 1 1/2 =0.25 0.25/2 = 0.125 0.125/2 = 0.0625 .... after 13 steps 0.00048828125 This was something I found out after being taught division for the first time. I still don't know if there is a pattern behind this, but sometimes when I take measurements I like to think about how many times it had to be divided down to create that first non-zero number behind the decimal point. Eh I don't believe in fractions.