Hello my good sirs and ladies. I shall report that the realm of Big-Oh is not as opaque as it at first appeared. Another course of mine that deals more heavily in actual computer-science programming techniques has introduced the ideas of sorting and selecting types of codes in accordance with their performance. This performance is measured through worst-case scenarios, which has consequently made understanding how to derive crude mathematical equations that describe the steps taken in code clearer. For example, binary searching--which involves breaking sorted lists of data into smaller parts and then reducing the size of the relevant part of the list until you have found the specific data you asked for--is based off of logarithms. In a sorted list of 2048 indices, binary searching would take approximately 11 steps in the worst case scenario (the item is not within the list) to complete the search, while a linear search pattern (looking at each index one by one) would take 2048 steps to complete the same search. As you probably know, the log(2048) with base 2 is equivalent to 11, and thus the general pattern of binary searches are: log(list_size) with base 2 = number of steps n. Big-Oh cases simply aim to state this process in a logical proof.
Alright, aside from that, I have been working on and off on the treacherous Last Theorem, and have managed--I think--to have proven the Last Theorem for all integer exponents that have a factor of 4. If you do not recall, the Last Theorem states that there are no three positive integers a, b, and c that can satisfy the equation a^n+b^n=c^n for any integer value of n greater than 2. In my uploaded photo, you will notice that I did not include a proof for all n that have a factor of 4, because--aside from the fact that it is intuitive to believe that what does not work for the number 4 will not work for the number 4(2) or 4(3) or 4(4) or... 4(n)--I at one point was frustrated by my inability to prove the theorem when n = 3, and ended up looking to the World Wide Web for guidance.
Feel free to peruse my effort and point out my mistakes. My problem-solving process was to transform the theorem into a mathematical expression, analyze the contrapositive and negation to determine what seemed easiest, and attempt a proof for a small portion of the theorem. My proof was a proof by negation. Hopefully, after careful review and anyone else's insight(s), I will be able to revise and/or expand this proof.
Thank you for reading and perusing!
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