Resolution: All whole numbers are interesting.

I apologize for the delay in my R3 argument. Trying to paste equations into DDO is frustratingly difficult. I would like to point out that the original structure for this debate asked both participants to use Round 2 for their opening statements. Con has provided both his opening statements and attack against my claims in his Round 2 post, putting me at a slight disadvantage. Therefore, in the interest of fairness, I think it would be reasonable to ask for a slight change in the format for the coming rounds: Round 2: Pro's opening statements, Con's opening statements and attack Round 3: Pro's attack and defense, Con's second attack and first defense Round 4: Pro's second attack and defense and closing statements, con's second defense and closing statements I hope Con will agree to these terms, as this debate has been incredibly fascinating to engage in thus far. Okay, let's begin the deconstruction! "4. In order for a number to be considered interesting, it must have a reason as to why it is interesting that can be expressed using words in the English language." I agree with the first three facts you make, but number four is misleading, as a number doesn't have to be interesting using English description. For example, 135 can be considered interesting since: [1] Also, there are literally an infinite number of ways to describe why a number is interesting using a combination of English and numbers anyway, which I will explain later using expository numbers. "If [the idea that primes are interesting because they have only two factors] was true then I could just say that even numbers are interesting because it is the most inclusive set of whole numbers divisible by 2, and odd numbers are interesting because it is the most inclusive set of whole numbers not divisible by 2." This is a false equivalency, but fair enough. If that doesn't satisfy your criteria of being considered "interesting," I can understand that. However, I would still argue the mystery behind prime numbers, the Ulam spiral, and finding the next highest prime number are all very interesting indeed. "I argue that not every number in that set can be interesting because they are not unique and have no interesting, non-trivial trait that separates it from the rest of the set." Okay, now we're getting somewhere. My opponent asserts the idea that since there are a finite amount of words in the English language and a finite amount of time to read a description as to why a number is interesting, it's impossible for all numbers to have a unique description about them. However, this is not necessarily true. I will now discuss a proof that literally every whole number has a unique property that makes it interesting. To begin, we have to explain what expository numbers are. I will be referencing source [2] for the majority of this argument. We will first look at all numbers that can be expressed by the sum of the kth power of the individual digits in the number and denote it: . In other words, you take the sum (s) of the kth power (k) of its digits (n). We can extend this idea a little further to get the following equation: , where "c" is a constant, and "a" is raising the original number to some power. Let me demonstrate the equation with an example using the number 666 as source [2] also indicates: Therefore, 666 is an expository number. In English, 666 is an expository number because: "666 is equal to the sum of the cube of the digits of its square plus the sum of the digits of its cube." It should, then, be easy to see that 666 is an interesting number, as not all numbers hold this property. But say you're not convinced. "What if 666 is not the only number for which this property holds true?" you ask. "It must have a unique property that no other numbers possess!" Fair enough. Again, referring to my second source, they do the math to see how many whole numbers are expository, and how many of them hold unique properties that no other expository number possesses. The author breaks all whole numbers up into one of three categories: 1. The number is not expository 2. The number is expository, but there exist other numbers that hold the same property, thus not making it a unique expository number. 3. The number is both expository and very likely to hold a unique property. Their findings were as follows: "Conjecture 1: The only positive integers of type (i) (i.e, the non-expository numbers) are:" [2] 11, 13, 14, 16, 19, 29, 37, 44, 55, 67, and 73. "Conjecture 2: The only positive integers of type (ii) are:" [2] 2, 3, 4, 5, 6, 7, 8, 9, 15, 17, 22, 23, 25, 26, 31, 38, 42, 47, 56, 59, 64, 76, 77, 79, 82, 86, 88, 89, 91, 92, 93, and 109 If these first two conjectures are in fact true, then all whole numbers must be interesting, as it should be trivial to find unique properties for the numbers listed above. All other numbers are both expository and have a unique property no other expository number possesses. For example: "135 is the only positive integer equal to the sum of the digits in its 4th power plus the sum of the digits in its 11th power." [2] I believe this satisfies Con's condition on when and how a number can be considered interesting. "When X is in the uninteresting category, it is interesting; when X is in the interesting category, it is uninteresting. Clearly, X does not belong in either category. It is a paradox. Therefore, my opponent's proof is invalid." This should support my idea that there are no uninteresting numbers moreso than Con's idea that some numbers are uninteresting. The fact it creates a paradox should demonstrate that the "Uninteresting" category was empty to begin with. If it wasn't, that's when the paradox occurs. I also apologize for the formatting of my R3 post. Trying to paste equations and images while making the post itself presentable and readable is quite a challenge. Maybe it's the computer I'm working on. I await Con's response. Thank you. Sources: [1]http://www2.stetson.edu... [2] http://www.cadaeic.net...