Resolution: All whole numbers are interesting.
Voting Style:  Open  Point System:  7 Point  
Started:  2/23/2017  Category:  Science  
Updated:  1 year ago  Status:  Post Voting Period  
Viewed:  946 times  Debate No:  100187 
Starting a relatively casualstyle debate for anyone interested. I am for the idea that every single whole number can be considered interesting, and my opponent will try to convince us the resolution is false. I will provide all necessary definitions I can think of. (Feel free to let me know if you want me to change any of the given definitions. I will consider reasonable alterations.) All: The whole quantity or extent of a particular group or thing. Whole number: All nonnegative integers, i.e. {0,1,2,3,...} Integer: A number that cannot be written without a fractional component, i.e. {...3,2,1,0,1,2,3,...} Interesting: Arousing curiosity or interest; holding or catching the attention (This is a difficult word to define in mathematical terms, as it can be subjective what is considered an "interesting" number, but that's the fun of this debtate!) There are four rounds to this debate. The general format will be as follows (though don't worry about adhering to the structure too strictly): Round 1: Acceptance Round 2: Opening Statements Round 3: Rebuttal/Attack Round 4: Defense and Closing Statements The only thing I ask is that you please try and see the debate through. I've had many forfeitures in my recent debates, and I don't want this one to go by the wayside. Happy debating to whoever accepts! Looking forward to this one.
I accept this challenge and will be taking the stance that not all whole numbers meet your definition of interesting. The round structure seems to suggest that I should not post any arguments now, so I will hold off. Thank you for posting this debate, I look forward to it 

I sincerely thank my opponent for accepting this debate, and wish them the best of luck in the coming rounds.
I will be making four claims to support my argument:
[C1] Proof
[C2] Interesting ≠ Practicality
[C3] Prime Numbers are Awesome
[C4] Other Ways of Identifying and Classifying Numbers
[Claim #1] Proof
You can actually explicitly prove (albeit in a slightlyhumorous fashion) that all whole numbers have to be interesting by using a proof by contradiction. A proof by contradiction is one where you assume the opposite of the proposition, and eventually find a contradiction, thereby proving the original proposition to hold true. The proof here is as follows:
Classify all numbers as being in one of two categories: Interesting an Uninteresting
My original proposition states all whole numbers are in the Interesting category, leaving the Uninteresting category empty.
Assume not
> Assume that there exists at least one number that is in the Uninteresting category.
> There exists one number that would then be considered the smallest uninteresting number, thereby making the number interesting as it has a unique quality that no other uninteresting numbers possess.
Therefore, that number cannot be uninteresting, and must be moved to the Interesting category.
This will inevitably lead to a contradiction, as you can repeat this process for every "Uninteresting" number.
∴ The original proposition holds: All whole numbers are interesting.
The crux of this argument is assuming that if a number holds a unique quality, you can (and should) classify it as interesting. Again, this is an ambiguous classification and is up to interpretation, but I am under the assumption that any reasonable person would consider a number like that to be considered "interesting". I can continue this line of logic further, but I will leave that to later rounds.
[Claim #2] Interesting ≠ Practicality
It would be quite difficult to say that every whole number has a practical application, and that it not what I am trying to argue. A number can be considered interesting and not necessarily have a realworld practical use. However, that also doesn't assume that we won't one day find an application for the number or its properties.
For example, the number 1894 can be expressed as 1^{4} 2^{3} 3^{2} 4^{1}. Interesting? I'd think so. Practical? Maybe not in most people's lives, but that doesn't discount the fact that this property is indeed both unique and interesting to 1894.
[Claim #3] Prime Numbers are Awesome
A prime number is one that has exactly two factors: 1 and itself. For example, the number 3 has only itself and 1 as factors, making it prime. If a whole number (above 2) is not prime, it must be composite, meaning it has more than 2 factors. For example, 9 has exactly three factors: 1, 3, and 9. 1 and 0 are neither prime nor composite, but they are obviously already both very interesting numbers to begin with. All prime numbers can be considered interesting for two reasons: The Ulam spiral and infinity. The Ulam spiral is a visual representation of all the integers written in a spiral pattern [1]. You then highlight every prime number. When you do this to a large enough degree, you will begin to see a slight pattern to the highlighted primes. To this day, mathematicians are still perplexed as to why this happen, and there is a lot of mystery surrounding this topic and why it happens, which would then lead us to believe that prime numbers are, in fact, very interesting.
There are also infinitelymany primes, which is another interesting fact. Mathematicians are constantly oneupping each other by running a computer program to try and find the next highest prime number. The last man to find the highest prime to date discovered one that was over 17 million digits long. a 17 milliondigit number that has only two factors is incredibly interesting! Not only that, he's also offering a $150,000 prize for the first person to find a prime over 100 million digits long [2].
[Claim #4] Other Ways of Identifying and Classifying Numbers
There are many, many other ways you can identify numbers besides just classifying them as "prime" or "composite" or "neither prime nor composite". For example, you have Mersenne primes, twin primes, sexy primes, "weird" numbers, "happy" numbers, "triangular" numbers, etc. with each one holding unique properties. With all of these crazy classifications, I would argue the BoP is on Con to show which numbers can be considered "uninteresting" and why.
I look foward to hearing your response Con.
Source(s):
First, I will list some seemingly trivial facts I think my opponent can agree with me on.


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, nontrivial 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 nonexpository 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: To clarify my fourth point in the previous round, "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," can still be applied to descriptions such as my opponent's reason of why 135 is interesting. He writes: In English, this would read, "The number 135 is interesting because it is the sum of its first digit, its second digit squared, and its third digit cubed." Because all numbers and math terms have verbal representations in the English language, I don't see why every possible reason to why a whole number is interesting can't be stated in the English language. Moreover, the reason does not really have to be in English. I only said that to simplify the argument. If, however, someone has a reason that is in French, I assume that the reason can be translated to English. "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 agree that the Ulam spiral is very interesting, especially because mathematicians don't know why this pattern arises. However, this does not mean that every prime number is interesting, and my opponent hasn't shown why each individual prime number is interesting (aside from his expository argument which I will get to). Next, my opponent used a source, http://www.cadaeic.net......, as evidence that all whole numbers are interesting by showing that many numbers have unique qualities regarding how they can be expressed. My opponent said that http://www.cadaeic.net...... shows that all but a few whole numbers can be shown to have a unique way to express it; therefore, all whole numbers are interesting. However, http://www.cadaeic.net...... actually shows something different; the study was actually only completed for all whole numbers below 1000. "Based on our examination of 10 through 1000, we make the following conjectures..." is a quote from http://www.cadaeic.net...... in regards to the three conjectures my opponent made. Rather then writing, "The only positive integers of type (i) (i.e, the nonexpository numbers) are," he should have written "The only positive integers less than or equal to 1000 of type (i)..." and so on. Therefore, this logic does not prove all whole numbers are interesting because it does not extend past 1000. However, even if the study did find the same results for all whole numbers above 1000 as well, this theory still does not account for my proof given in CLAIM 1 of my previous argument. Keeping in mind that numbers have finite representations in the English language, if this logic was extended to all whole numbers to infinity, there would eventually be a point where the numbers were so large it would be impossible to state the "interesting" numeric representation in a reasonable amount of time. In regards to my attack on my opponents Claim #1, my opponent writes, "The fact it creates a paradox should demonstrate that the 'Uninteresting' category was empty to begin with." This is incorrect. We are debating whether all primes numbers are interesting, not whether there are no uninteresting numbers. You might think that if there are no uninteresting numbers, all numbers must be interesting; however, my opponent's claim reached a paradox  two of the results found contradict each other. Therefore, no conclusion can ever be made from that claim. I will pose my last argument as a challenge. Consider the following arbitrary whole number: 76941309571390671361571395 If every whole number is interesting, then I challenge my opponent to find what is interesting about this number. I know that his ability to find an interesting trait will not definitively prove either of our viewpoints, but I think it will give us more insight to the problem. I also wanted to apologize for not following the round format on my second post. The new format should be fine, thank you. 

I sincerely thank my opponent for joining me in this debate. It has been a lot of fun for me, as I've had to do quite a bit of research to understand how exactly a number can be classified as "interesting." And learning about expository numbers has also been quite fun. On with my final statements! "...all numbers and math terms have verbal representations in the English language..." I'm obviously not disputing this statement, but your claim that because there is a finite amount of English words, there is a finite number of ways to describe the uniqueness of a number that differentiates it from all other numbers is fallacious. That's like saying there's a finite amount of whole numbers because after a certain point, we will run out of names for them (or it will be so large we won't have enough time in our lives to say the number). Just because it would take an arbitrarilylong amount of time to describe the uniqueness of a particularly large number doesn't prove that limits the uniqueness of the number itself. The fact you can represent the interesting qualities of a number numerically demonstrates this. Claim disputed. "...my opponent hasn't shown why each individual prime number is interesting..." I was simply making a point that there are many ways to classify numbers, each of which can be considered interesting. Obviously your idea that numbers have to have a unique property to be considered interesting would discredit the idea that prime numbers are interesting because they are prime, which I used to my advantage in my expository numbers proof. However, my expository number proof would, then, show that large prime numbers are interesting (not because they are prime, but because they have a unique and interesting representation in expository form). Point defended. "However, http://www.cadaeic.net......... actually shows something different; the study was actually only completed for all whole numbers below 1000." I encourage Con to go back and read the article again: "These results suggest that every integer > 999 is an expository number in hundreds of thousands of ways, or more. Surely it will be possible to find one or two of these whose intersection of solutions sets is just the value n. Note that we have deliberately limited ourselves to the intersection of two sets. We could take three or more, if necessary, in order to obtain a property that n uniquely satisfies, but we conjecture that this will never be needed. [1]" They stopped at 1,000 because it was unnecessary to continue. Every number in the range of 900999 has over 145,000 different variations of being an expository number, and with enough time and research, there would be enough intersections between those variations that would give each number a unique expository property. Refer back to my 135 example in Round 2. The proof suggests very, VERY strongly that every number above 1,000 is uniquely expository. Point defended. "We are debating whether all primes numbers are interesting, not whether there are no uninteresting numbers...my opponent's claim reached a paradox  two of the results found contradict each other. Therefore, no conclusion can ever be made from that claim." First, my proof by contradiction was not just talking about primes, but all whole numbers. Second, the fact there's a contradiction means the original proposition in the proof is untrue. Namely, that there are no uninteresting numbers. I feel there may have been a miscommunication there. Point defended. "I challenge my opponent to find what is interesting about [the number 76941309571390671361571395]." This is an argumentum ad ignorantiam [2], and thus is irrelevant. Let's recap:
With all of this information to consider, it should be clear what the decision in this debate is. Vote Pro. Again, a final thank you to Con, and I look forward to hearing his final defense and closing statements. Sources:
"That's like saying there's a finite amount of whole numbers because after a certain point, we will run out of names for them (or it will be so large we won't have enough time in our lives to say the number)." That is exactly what I am saying, except in regards to the reason to why a number is interesting. My opponent's definition of interesting requires a human to have curiosity aroused. As I explained before, humans cannot possibly have curiosity in all whole numbers as humans are finite beings. Claim defended. I want to thank my opponent for this debate, and I especially thank any reader who made it this far in the debate. I hope you enjoyed it. Throughout the debate, I explained how humans cannot have interest in all infinite amount of whole numbers, and my opponent was unable to provide sufficient evidence to dispute me. Last round, I challenged my opponent to find something interesting about the number 76941309571390671361571395. My opponent was either unable, or you might say "uninterested", in finding anything interesting about that number. To the reader I ask, do you think there is something interesting about the number 76941309571390671361571395? What about the number 91 or 5670 or 420 or 45679936266153577389007362728197577388299846648489396362748483939? Do you think every number could possibly be interesting? If yes vote pro. Otherwise, give me your support. To close I want to make something clear. Mathematics is the coolest thing that ever existed. It is actually my major at college. Many numbers are interesting. I am surprised my opponent left it at the Ulam spiral. He could have introduced topics even more interesting than that one. I merely understand the vastness of infinity. 
>Reported vote: KnightOfDarkness// Mod action: Removed<
3 points to Con (Arguments), 2 points to Pro (Sources). Reasons for voting decision: RFD in comments
[*Reason for removal*] Sources are insufficiently explained. The voter points to a single source made in the debate, states it"s trustworthy and explains why, but never compares that source (or any of the other sources from Pro) to Con"s sources.
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 91 is the smallest pseudoprime in base 3.
 5670 is a value of n for which `6;(n) and `3;(n) are square.
 You should already know why 420 is interesting. Also, it's the smallest number divisible by 1 through 7.
http://www2.stetson.edu...
In my Round 3 post, I use expository numbers to prove that literally all numbers are interesting. Con's argument that English is finite and therefore some numbers would be interesting for the same reason is not true, which I stated in the debate, since there are an infinite number of ways to classify each number using English. If each number past 109 is uniquelyexpository (which evidence would strongly suggest that to be the case), that would demonstrate this idea very easily.
Arguments:
I voted con because mainly the argument that there is a finite amount of ways to express ideas in the English language, but an infinite amount of numbers, thus many of the numbers would have the same reason for being interesting, thus are not, was compelling. Pro tried to defend against this by calling it fallacious, but this seemed to be a weak argument because they didn't offer what specific logical fallacy it is, and it seems unclear as to how it is one. Con's argument against pro's argument for numbers being interesting through the proof by contradiction, and claiming it is a paradox, and thus would leave some numbers being interesting and uninteresting at the same time was also a compelling argument. If it's in some middle state, it would be neither interesting nor uninteresting, and pro needed to prove all numbers are interesting
Sources:
I voted pro because they offered one source from a university, and universities are generally trustworthy sources for information, as what they publish tend to be from experts in the field. In general, all of their sources were with authors with a background expetise in Math