This note has stopped at 300, and we are now finding the most boring number.

Congrats! There have been so many comments on the first note, that it's overloading the system. As such, I have locked the first post and we will continue on here.

The point of this note is to list out an interesting property for each positive integer. Reply to the largest number N, and state why N+1 is interesting in 14 words or less.

**Rules:**

1. Start with "N is ...".

2. Make sure you use 14 words or less.

3. Do not reply out of sequence.

4. Do not reply to your own comment. (Applicable to 9 onwards)

Proposition: Every integer has an interesting property that can be described in 19 words or less.

Proof by contradiction: Suppose that there exists numbers which do not have an interesting property. Let \(S\) be the smallest of these numbers by the Well-Ordering Principle. Then,

"S is the smallest integer that cannot be described in 14 words or less."

which is a contradiction.

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## Comments

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TopNewest\(\Large \text{Enlist all interesting properties of }\color{blue}{2016}\)

Reply to this thread :)

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\(2016^{2} + 2016^{3} = 8197604352\), a number that contains one of each digit.

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\({2}^{5}+{2}^{6}+{2}^{7}+{2}^{8}+{2}^{9}+{2}^{10}=2016\)

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From Brian Charlesworth: 170 is the smallest number \(n\) such that both \(\phi(n)\) and \(\sigma(n)\) are perfect squares.

170 is the maximum possible check-out score in darts.

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171 is the smallest nontrivial odd triangular number that is also tridecagonal

@Calvin Lin What are the guidelines for the length of comment, especially if it contains numbers or symbols?

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172 is the smallest integer that is a repdigit (repeated digit) in 4 bases.

Base 6, 42, 85, 171.

Single digit numbers are not considered repdigits.

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(The bracketed phrase can be deleted if necessary to comply with the 14-word limit.)

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@Michael Mendrin We have a couple of parallel threads going here, so I figure we should streamline on this one.

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Let's see if you can do it without mentioning the Declaration of Independence.

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176 is the number of possible partitions of the number 15.

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\({ 179 }^{ 2 }=32041\)

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(Strobogrammatic means it looks the same reversed left-right, or up-down)

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(Scroll all the way to the bottom to see what the latest number is).

@Michael Mendrin Ideally, it should be 14 words or less (verbally). I haven't been strictly enforcing this rule, because by the time I realize that (say) #123 doesn't work, we're already working on #134.

So yes, #170 (part 1) doesn't fit with this ruling.

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Well, what keeps me going with this thread is, "are we ever going to finally come to a truly boring number nothing special can be said about it, without a ton of qualifiers?" And then that will be the special quality of that number, being the first of such.

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Yup! That's one thing I'm looking for.

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\({ 1 }^{ 3 }+7^{ 3 }+{ 3 }^{ 3 }=371\)

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That's a pretty interesting fact to come across. Only prime? Hm....

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174 is the sum of consecutive integers 5, 6, 7, 8

eh

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@Brian Charlesworth In relation to "without a ton of qualifiers", if I recall correctly, there is a result in Number Theory, which states that any integer \(n\) can be uniquely defined as the sum of \( a_i \) positive distinct \(b_i \) powers in \( c_i \) ways, for some set of constants \( a_i, b_i , c_i \).

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The interesting question is, given that in English it requires a certain number of words to actually state a [large] number, can it be described by its properties with fewer words? Likewise, can it be generally more efficient to describe a [large} number by how it may be a sum of powers in so many ways?

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@Brian Charlesworth No, it will require more triples, because \( a^2 \times 174 \) can also be represented as the sum of 4 perfect squares in 6 ways. An example of a potential triple to add is \( (174, 123, 1) \)? But I don't recall if there were restrictions on these values (like having \( a_i < n \) or \( c > 1 \) ).

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The theorem Calvin mentioned seems like a considerably more formidable one to prove.

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@Brian Charlesworth There is an obvious reason why "there is a number whose squares start with four (or n) identical digits. Hint: The gap in consecutive squares is approximately \( \sqrt{n} \).

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This is in reply to what number?

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Brian's comment

My claim is that "It is obvious there is a (infinitely many) number whose square starts with four (or n) identical digits." It is also not too hard to find out what the smallest one is, and in fact I believe that the smallest answer would have (close to) \(n\) digits. See this problem.

In the case of 4 digits, \( 3334^2 = 11115556 \) would be the smallest example.

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