Every Integer is Interesting - Part 2

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.

#NumberTheory

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Comments

$\Large \text{Enlist all interesting properties of }\color{#3D99F6}{2016}$

Reply to this thread :)

Nihar Mahajan - 5 years, 5 months ago

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

Brian Charlesworth - 5 years, 5 months ago

${2}^{5}+{2}^{6}+{2}^{7}+{2}^{8}+{2}^{9}+{2}^{10}=2016$

Michael Mendrin - 5 years, 5 months ago

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.

Calvin Lin Staff - 5 years, 10 months ago

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?

Michael Mendrin - 5 years, 10 months ago

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.

Calvin Lin Staff - 5 years, 10 months ago

@Calvin Lin – Element 173 is thought to be the highest possible element.

Alex Li - 5 years, 10 months ago

@Alex Li – 174 is (the least number) expressible as the sum of 4 positive distinct squares in 6 ways.

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

Brian Charlesworth - 5 years, 10 months ago

@Brian Charlesworth – 175 is the least $n \gt 1$ such that $n^{6} + 6$ is prime.

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

Brian Charlesworth - 5 years, 10 months ago

@Brian Charlesworth – Brian, you're welcome to tackle 176

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

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – Hahaha I was going to wait until we get to 1776 before doing that. :)

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

Brian Charlesworth - 5 years, 10 months ago

@Brian Charlesworth – 177 is the smallest magic constant for a 3x3 prime magic square

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – 178 is a digitally balanced number, as its binary number (10110010) has an equal number of zeros and ones.

Nihar Mahajan - 5 years, 10 months ago

@Nihar Mahajan – 179 is the smallest prime which square is a cyclops number

${ 179 }^{ 2 }=32041$

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – 180 is the sum of the interior angles of a triangle in Euclidean space.

Brian Charlesworth - 5 years, 10 months ago

@Brian Charlesworth – 181 is the only three digit non-cyclops strobogrammatic prime

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

Michael Mendrin - 5 years, 10 months ago

@Alex Li – Also, 173 is a Cuban prime, i.e., a prime number which is the difference between two consecutive cubes.

Michael Mendrin - 5 years, 10 months ago

(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.

Calvin Lin Staff - 5 years, 10 months ago

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.

Michael Mendrin - 5 years, 10 months ago

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

Calvin Lin Staff - 5 years, 10 months ago

@Calvin Lin – 173 is only prime which sum of cubed digits is same reversed

${ 1 }^{ 3 }+7^{ 3 }+{ 3 }^{ 3 }=371$

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – Wrong comment to reply to.

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

Calvin Lin Staff - 5 years, 10 months ago

@Calvin Lin – I didn't see Alex's contribution. Okay....so

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

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – @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 $a_i, b_i , c_i$ .

Calvin Lin Staff - 5 years, 10 months ago

@Calvin Lin – I also recall another result in Number Theory which states that any integer can be uniquely defined as a sum of integer multiples ${ a }_{ i }$ of integer powers ${ b }^{ i }$ .

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?

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – Hm, that might be the version that I recall. Can't really find the statement though.

@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 $c > 1$ ).

Calvin Lin Staff - 5 years, 10 months ago

@Calvin Lin – O.k., I'll keep an eye out for such a theorem and any potential conditions. That is one powerful result if it is indeed the case.

Brian Charlesworth - 5 years, 10 months ago

@Calvin Lin – Huh, I don't know that one. Is it connected to the Hilbert-Waring theorem? So would that mean that $174$ is "defined" by the triple $(4,2,6)$ ? While $174$ is the smallest such number for which this triple applies, I assumed that there would be other such integers as well.

Brian Charlesworth - 5 years, 10 months ago

@Brian Charlesworth – I thought Waring's problem addressed the question of how many equal powers of distinct integers it takes to equal to any integer? For example, with up to 143 distinct 7th powers of integers, any integer can equal to a sum of them.

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

Michael Mendrin - 5 years, 10 months ago

@Michael Mendrin – You're right. I was just trying to connect the theorem Calvin mentioned to something familiar and Waring's problem was what first came to mind. Given Calvin's comment below the theorem may come with some conditions, so I'll need to do a bit more research.

Brian Charlesworth - 5 years, 10 months ago

@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 $\sqrt{n}$ .

Calvin Lin Staff - 5 years, 10 months ago

This is in reply to what number?

Michael Mendrin - 5 years, 10 months ago

Brian's comment

Haha $88804 \div 4 = 22201,$ so given that $298$ is the second, $149$ must be the first. The third such number is $334^{2} = 111556.$ (I guess the next question is to find out if there are any numbers whose squares start with four identical digits. I did a quick search up to 1000 without success.)

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.

Calvin Lin Staff - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$