User:Antony9595

I am Antony T H Fung, an upper sixth Maths student from Colchester Royal Grammar School.

If you found this page, probably it is because you saw the term "Antony Numbers" and had no idea what it means, so you googled it.

Well, few months ago, when I was playing with my beloved calculator, I was wondering if I enter a real number a little bit larger than 1 into my calculator and press "Ans^Ans", how much time do I have to press "=" before I get "MATH ERROR". To study this, I defined a function (which I haven't give a proper name to it). Lets call the function "Antony Function" for the sake of making it easier to explain (I know it sounds very arrogant, so this is just a temporary name, I am not using this name for it forever).

Antony: positive real numvers-->non-negative integers Antony(x)=minimum value of k such that $$a_k$$>2, where $$a_0$$=1+1/x and a_(n+1)=$$a_n$$^$$a_n$$ for all non-negative integers n.

Then I started to find the values for Antony(x):

Antony(1)=1

Antony(2)=2

Antony(3)=3

Antony(4)=4

Antony(5)=5

Antony(6)=6

Antony(7)=7

Antony(8)=8

Antony(9)=9

Antony(10)=10

Antony(11)=12

...

Wow...Antony(x) is very close to x. Wait a minute, 11 is missing at the right hand side.

To study those missing values, I defined a set of numbers:

y belongs to the set of numbers if and only if it satisfies both conditions:

1. y is a positive integer 2. there doesn't exist an positive integer x such that Antony(x)=y

Then, I started to find the elements of the set.

Result: 11,86,628,4627,34182,252599,...(I can only find these because of the limit of the accuracy in my C++ program)

I spotted that 11,86,628,4627,34182,252599,... is a very (but not exactly) geometrical sequence.

(I know why Antony(x) is so close to x by looking at the Taylor series, and by using Taylor series and many times of Mathematical Induction, I proved that for all positive integers x, Antony(x+1)-Antony(x)>=1. But I still have absolutely no idea why the series is so geometrical.)

Approximately a month after studying these stuffs, teachers in my school started talking about EPQ, then I suddenly realized that I can use it as an EPQ project. Then the problem came: what should be the topic of the EPQ project?

To solve this problem, I named the set of numbers {11,86,628,3427,34182,252599,...} "Antony Numbers" and use this name as my EPQ project title. (I know this name sounds arrogant, but I have to put something on my EPQ project title...anyway, I may change this name after submitting my EPQ)