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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.

Note by Calvin Lin
1 year, 8 months ago

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

Reply to this thread :) Nihar Mahajan · 1 year, 2 months ago

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@Nihar Mahajan \(2016^{2} + 2016^{3} = 8197604352\), a number that contains one of each digit. Brian Charlesworth · 1 year, 2 months ago

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@Nihar Mahajan \({2}^{5}+{2}^{6}+{2}^{7}+{2}^{8}+{2}^{9}+{2}^{10}=2016\) Michael Mendrin · 1 year, 2 months ago

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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 \( \sqrt{n} \). Calvin Lin Staff · 1 year, 7 months ago

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@Calvin Lin This is in reply to what number? Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 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 · 1 year, 7 months ago

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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. Calvin Lin Staff · 1 year, 8 months ago

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@Calvin Lin 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 · 1 year, 8 months ago

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@Michael Mendrin Yup! That's one thing I'm looking for. Calvin Lin Staff · 1 year, 8 months ago

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

\({ 1 }^{ 3 }+7^{ 3 }+{ 3 }^{ 3 }=371\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Wrong comment to reply to.

That's a pretty interesting fact to come across. Only prime? Hm.... Calvin Lin Staff · 1 year, 8 months ago

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

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

eh Michael Mendrin · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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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. Calvin Lin Staff · 1 year, 8 months ago

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@Calvin Lin 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 · 1 year, 8 months ago

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@Michael Mendrin 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 · 1 year, 8 months ago

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@Calvin Lin Element 173 is thought to be the highest possible element. Alex Li · 1 year, 8 months ago

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@Alex Li Also, 173 is a Cuban prime, i.e., a prime number which is the difference between two consecutive cubes. Michael Mendrin · 1 year, 7 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@Brian Charlesworth 177 is the smallest magic constant for a 3x3 prime magic square Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 178 is a digitally balanced number, as its binary number (10110010) has an equal number of zeros and ones. Nihar Mahajan · 1 year, 8 months ago

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

\({ 179 }^{ 2 }=32041\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 180 is the sum of the interior angles of a triangle in Euclidean space. Brian Charlesworth · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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Comment deleted Jul 24, 2015

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@Nihar Mahajan 182 is the first pronic sphenic number not divisible by 10 Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 183 is the smallest \(n\) such that \(n\) concatenated with \(n + 1\) is a square. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 184 seems like a pretty good candidate for a "number of no interesting account". Why? Because

It is not a prime (nor semiprime)
It is not a pronic number (nor sphenic number)
It is not a happy number (nor lucky number)
It is not a perfect square (nor any other power)
It is not a factorial (nor double factorial nor sub factorial)
It is not a Fibonacci number (nor Lucas number)
It is not a Triangular number (nor any other small order polygonal nor figurate number)
It is not a Catalan number (nor Delannoy number)
It is not a Palindrome (nor Cyclops number)

It doesn't look like anything if you reversed it in a mirror or turned it upside-down or any of that

It is a DEFICIENT and EVIL number

Well, but we can say this about 184:

184 is the maximum number of areas 14 circles can divide a plane into Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 185 is even more of a challenge. It is odious and deficient, but there seems to be no unique characteristic other than the following:

185 is the number of conjugacy classes in the automorphism group of the 8-dimensional hypercube. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Even though that is a jargon-filled multisyllabic jawbreaker, that IS a pretty unique position for any number to have, this thing about 8 dimensional hypercube. I bet you that as a consequence of this, some universe exists somewhere.

Meanwhile, time for me to go to sleep. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin I see that you have turned 65. When was your birthday? Nihar Mahajan · 1 year, 8 months ago

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@Nihar Mahajan As noted in Every Integer..., the \(19th\) of this month (July). See \(19\) in this list. Was gone the whole weekend to have fun. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Oh , I see. Nihar Mahajan · 1 year, 8 months ago

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@Michael Mendrin Hahahaha Well, o.k., I suppose that does make it special, then. :) And for another jawbreaker....

186 is the number of degree 11 irreducible polynomials over GF(2).

(GF(2) is a Galois field of 2 elements.) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 187 is the area of a standard 11 x 17 sheet of paper.

Also 187 is the California Penal Code for murder. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 188 is the number of nonisomorphic semigroups of order 4. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth The product of two pairs of twin primes with a gap of 4 always ends in 189

E.g.

\(821\cdot 823\cdot 827\cdot 829=463236778189\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 190 is the largest number such that it and its prime factors are Roman palindromes.

(A Roman palindrome is an integer that is palindromic when written in Roman numerals.) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Also, there are only 190 polycubes of order 4. No boring number nomination for this one. Michael Mendrin · 1 year, 7 months ago

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@Brian Charlesworth The 191 orientable octahedral manifolds (whatever that means)

The 191 orientable...

Also, $1.00 + $0.50 + $0.25 + $0.10 + $0.05 + $0.01 in US coins = $1.91 Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Yikes, that's daunting. However, I suppose that there is yet another universe out there that depends on this fact. I was thinking something more mundane: 191 is the sum of the standard US coin denominations. Anyway ....

192 is the smallest number with 14 divisors. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 193, when divided by 71, gives the best approximation of e using 5 digits or fewer Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin The sum of the proper divisors of 194 is 100. Alex Li · 1 year, 8 months ago

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@Alex Li There are 195 sovereign countries in the world. Alex Li · 1 year, 8 months ago

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@Alex Li It is not known whether the algorithm of reversing the digits and adding it to the number itself will ever reach a palindrome if the starting number is 196. Alex Li · 1 year, 8 months ago

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@Alex Li 197 is the only prime 3-digit Keith number. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Also Circular Prime and Super Catalan. No boring number here. Michael Mendrin · 1 year, 7 months ago

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@Brian Charlesworth 198 is one of the most commonly posted prices in stores Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Also Pell-Lucas number. Not boring. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 199 is the only 3-digit Lucas number that is also a permutable prime. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 200 is the smallest number which cannot be changed into a prime by changing one digit (orly!) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 201 reversed is 102 Brew, brewed in downtown Los Angeles during noir days

102 Brew

102 Brew

Looking southbound on Santa Ana Freeway next to downtown Los Angeles, circa 1957.

Otherwise, 201 is the smallest pentadecagonal and icosagonal number. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 202 is the number of partitions of 32 into powers of two.

\(202 = (2 + 3 + 5 + 7)^{2} - (2^{2} + 3^{2} + 5^{2} + 7^{2}).\) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth The product of the divisors \( 1\cdot 7\cdot 29=203 \), where \(1729\) is the Hardy-Ramanujan Number Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 204 is the number of different squares on a chessboard. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth There are 205 twin primes less than 10,000 Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Also both Primitive Sequence number and Wolstenholme number. No nomination here. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 206 is the number of bones in a typical adult human body. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 207 is the smallest sum of primes using digits 1 through 9 once

\(2+5+7+43+61+89=207\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 208 is the number of 5-bead necklaces using beads of 4 colors. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 209 is the smallest that can be expressed as a sum of three squares in six different ways Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 210 is the product of the first 4 primes, (and hence the only 3-digit primorial). Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 211 is the largest known prime number that is not a sum of a prime and a triangular number Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin \(212^{\circ}\) Fahrenheit is the boiling point of water at sea level. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth \(213^{2}=1!+2!+3!+7!+8!\) Julian Poon · 1 year, 8 months ago

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@Julian Poon 214 is the minimum total volume of three different strictly oblong bricks

\(1\cdot 8\cdot 9+2\cdot 5\cdot 7+3\cdot 4\cdot 6=214\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 215 is the number ordered positive integer 4-tuples \((a,b,c,d)\) such that

\(\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} + \dfrac{1}{d} = 1.\) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Of COURSE 216 is a classic. This is something even ancient Greeks observed:

216 is the smallest cube which is the sum of three cubes

\({ 3 }^{ 3 }+{ 4 }^{ 3 }+{ 5 }^{ 3 }={ 6 }^{ 3 }=216\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 217 is the sum of the positive divisors of 100, i.e., \(\sigma_{1}(100) = 217.\) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 218 is the number of distinct ways to color the edges of a cube with two colors

You know, Brian, I think these number facts offer possibilities as Brilliant.org problems. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Yup, I started creating some called "Interesting X". But of course, we could just ask the direct version like:

How many ordered positive quadruples \( (a, b, c, d) \) are there to \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = 1 \).

Seems like a somewhat painful counting / case checking though. Calvin Lin Staff · 1 year, 7 months ago

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@Michael Mendrin Yes, I suppose they do. I was waiting until we found a truly boring number and then designing a question that had it as the answer, just to make it interesting. Looking ahead, 224 looks like a good candidate, and if no one else jumps in it will be your challenge to prove me wrong.. But for now .....

219 is the smallest number expressible as a sum of four positive cubes in two ways. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth This one is a bit notable:

220 is the largest difference between two consecutive primes for all primes < 100,000,000 Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Also, only the 4th smallest power of 2 that contains the digits 666. Too cool to be boring. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Huh, I'll have to keep the prime gap table in mind for future numbers. This next one is a personal favourite...

221(B) Baker Street, London, England, is the (fictional) address of Sherlock Holmes. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 222 is smallest three digit number with prime only digits (!) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 223 is a prime island and the least prime whose adjacent primes differ by 16. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 224 is the smallest Mendrin-Hardy-Har number that is the sum of two cubes

\({ 2 }^{ 5 }\cdot 7=224={ 2 }^{ 3 }+{ 6 }^{ 3 }\)

where a Mendrin-Hardy-Har number is a number of the form \( { p }^{ q }\cdot r \) where \(p, q, r\) are primes such that \(p+q=r\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 225 is the smallest number that is a polygonal number in five different ways.

225 is the smallest square to have one of every digit in some base (3201 in base 4). Pranshu Gaba · 1 year, 8 months ago

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@Pranshu Gaba At most 226 different permutation patterns can occur within a single 9-element permutation. Nihar Mahajan · 1 year, 8 months ago

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@Nihar Mahajan The 227th harmonic number is the first to exceed six. Pranshu Gaba · 1 year, 8 months ago

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@Pranshu Gaba There are 228 matchings in a ladder graph with five rungs Nihar Mahajan · 1 year, 8 months ago

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@Nihar Mahajan 229: Smallest prime, when added to the reverse of its decimal representation, yields another prime (229 + 922 = 1151) Pranshu Gaba · 1 year, 8 months ago

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@Pranshu Gaba There are 230 unique space groups describing all possible crystal symmetries. Nihar Mahajan · 1 year, 8 months ago

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@Nihar Mahajan 231 is the only known number greater than six that is both hexagonal and octahedral

One US gallon is exactly 231 cubic inches. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin @Michael Mendrin Well played with the Mendrin-Hardy-Har number. :)

232 is the number of symmetric permutation matrices. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 233 is the only known Fibonacci prime which sum of digits is a Fibonacci number

Brian, I have no idea why the notable Mendrin-Hardy-Har numbers is not listed in OEIS. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin I'm trying to figure out what the next Mendrin-Hardy-Har number is, (or if there even is one). In the meantime ....

234 is a postage stamp problem solution for 4 denominations and 8 stamps. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 235 is the smallest non-trivial number that is both heptagonal and centered triangular

Here's how the Mendrin-Hardy-Har sequence goes

\(40, 45, 175, 224, 1573, 5491, 26071, 26624, 72283,...\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin Oh, sorry, I guess I meant the next Mendrin-Hardy-Har number that was also a sum of two cubes.

236 is the number of different connected graphs with 8 vertices and 9 edges.

The product of the digits of 236 is the reverse of the sum of its prime factors. (\(2*3*6 = 36, 2 + 2 + 59 = 63.\)) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Oh, well, yes, the next such Mendrin-Hardy-Har number that is the sum of two cube is a really big number. I'll see if I can find any at all.

237 is the smallest number such that the first three multiples of it has the digit 7 Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 238 is the smallest untouchable number that is the sum of the first \(n\) primes for any \(n \gt 2.\)

(2914 is the next such untouchable number. Question to consider: Is the set of all untouchable numbers that are also prime sums infinite? Erdös proved that the set of untouchable numbers is infinite, so the question is reasonable to ask.) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Machin's 1706 formula for \(\pi\)

\(\dfrac { \pi }{ 4 } =4ArcTan\left( \dfrac { 1 }{ 5 } \right) -ArcTan\left( \dfrac { 1 }{ 239 } \right) \)

Also, 239 is the largest number that cannot be the sum of 8 cubes or less. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 240 is the maximum number of divisors for any integer less than 1,000,000.

Also, \(n^{m} - n^{m-4}\) is divisible by 240 for any integer \(m \gt 7, n \in \mathbb{Z^{+} \cup \{0\}}.\) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 241 is

a) the smallest prime such that adding it to its reverse results in a palindromic prime
b) the smallest non-palindromic prime that is the sum of a number and its reverse

Regarding 240, I wonder what would happen if there was 240 degrees in a circle? Then every divisor would represent a constructible regular polygon, and all trigonometric quantities of such angles would be algebraic. Who's idea was it to go with 360 degrees? Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin I think we have the ancient Babylonians to blame for the 360 degree scale; too bad they weren't better mathematicians, since a 240 degree scale would have made much more sense.

242 is the smallest \(n\) such that \(n, n + 1, n + 2\) and \(n + 3\) have the same number of divisors. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth Brian sir , don't you sleep? :P

243 is the largest 3-digit number that is a fifth power \((3^5)\). Nihar Mahajan · 1 year, 8 months ago

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@Nihar Mahajan 244 is the smallest non-trivial number that is both the sum of two squares and the sum of two fifth powers. Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 245 is the greatest 3-digit number \(n\) that divides the (left) concatenation of all numbers \(\le n\) written in base 2. Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 246 is the only number with exactly three prime factors that uses digits 1, 2, 3, 4 once.

\(246=2\cdot 3\cdot 41\) Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 247 is the smallest possible difference between two integers that together contain each digit exactly once.

(\(247 = 50123 - 49876.)\) Brian Charlesworth · 1 year, 8 months ago

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@Brian Charlesworth 248 is the smallest hexanacci number with digits in geometric progression Michael Mendrin · 1 year, 8 months ago

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@Michael Mendrin 249 is a major highway in Texas. Alex Li · 1 year, 7 months ago

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@Alex Li 250 is the number of integer solutions to the equation \(\dfrac { 1 }{ a } +\dfrac { 1 }{ b } +\dfrac { 1 }{ c } =\dfrac { 1 }{ 5 } \)

Sure major Texan highway, it goes all the way to Tomball and beyond out from Houston. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Pausing for the moment, I regard 249 as a contender for the most boring number thus far. The best I could come come up with was:

249 is the least \(n\) such that \(8^{n} \pmod{n}\) is \(14.\)

Before going further, I'd like to find out if either of the presented 'facts' about 249 thus far are sufficiently interesting to move on, or if not, if anyone can come up with a new fact that is sufficiently interesting. Otherwise, we might just have a winner in the boring contest, in which case 249 is officially the smallest boring number, which will then be its claim to fame, (as oxymoronic as that may be). Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 249 is a good candidate for the most boring number, up to 250. Only in trivial cases is it a figurate number, and there's a lot of figurate numbers. But it is a famous Texas highway! Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Well, yes, there is that, and it also a famous highway in la belle province of Quebec, (passing through the township of Asbestos, no less), as well as a National Route in Japan, (from Nanao to Kanazawa). Also, as an Angel Number, 249 is a message from your (inner and outer) angels that you have been receiving intuitive and angelic guidance about your life purpose and soul mission.

In other words, I nominate 249 as the most boring number up to 250. Any seconders? Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth Let me first carefully review past submissions. No hurry with this one.

Edit: Actually, 249 does have this interesting property

\({ 249 }^{ 3 }=15438249\)

so it fails to achieve the status of the "most boring number". Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Well, o.k., I suppose I'll let 249 scrape by, then. :) Brian Charlesworth · 1 year, 7 months ago

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@Michael Mendrin O.k., yes, there's no hurry. Moving on ....

251 is the smallest number expressible as the sum of 3 cubes in two distinct ways.

Also, 251 is the number of square submatrices of a any 5 x 5 matrix. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth \(\dfrac { 10\cdot 9\cdot 8\cdot 7\cdot 6 }{ 5\cdot 4\cdot 3\cdot 2\cdot 1 } =252\) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 253 is the smallest non-trivial triangular star number. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 254 is the first nonzero number of the form \({ n }^{ 8 }-n!\) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 255 is the greatest integer that can be represented as an 8-digit binary number. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 256....um...? This one bytes.

Okay, the Pisano period for the Fibonacci series in base \(256\) is \(384\), which is \(3\) times \(128\), and the Pisano period for same in base \(128-1=127\) is \(256\). Eh... This is a remarkably forgettable factoid.

Well, no, maybe not so forgettable. Let me have another look at that. See this problem

Fibonacci... Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Hahaha Yeah, 256 is a special number in many ways. :)

257 is the only 3-digit Fermat prime.

257 is the smallest number that is the sum of two distinct positive 8th powers. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 258 is the smallest magic constant for a 4x4 magic square using 16 consecutive primes. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin This is a stretch ....

259 is the only 3-digit integer \(n\) such that \(n\) divides the (right) concatenation of all positive integers \(\le n\) written in base \(25.\)

259 is the only 3-digit number of the form \(\dfrac{6^{k} - 1}{5}\) for some non-negative integer \(k.\) Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 260 is the magic constant in a 8x8 magic square using numbers 1 through 64 Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 261 is the number of different ways to dissect a hexadecagon, (16-gon), into 7 quadrilaterals. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 262 different polykites can be formed from 7 kites Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 263 is an irregular prime , since it divides the numerator of the Bernoulli number \(B_{100}\). Nihar Mahajan · 1 year, 7 months ago

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@Nihar Mahajan 264 is the largest known number which square consists of alternating digits 69696 Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 265 is the number of derangements of 6 elements. Nihar Mahajan · 1 year, 7 months ago

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@Nihar Mahajan \({ 2 }^{ { 2 }^{ 3 } }+{ 2 }^{ 3 }+2=266\) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 267 is the number of groups of order 64. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 268 is the smallest number which product of its digits is six times the sum of its digits Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 269 is the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix. Nihar Mahajan · 1 year, 7 months ago

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@Nihar Mahajan 270 is the smallest positive integer that has divisors ending in each of the digits \(1\) through \(9.\)

Owls are able to rotate their heads a full \(270^{\circ}.\) Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 271 is the first three digits of Euler's constant Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 272 is a Euler zigzag number. Alex Li · 1 year, 7 months ago

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@Alex Li -273 is Kelvin 0 Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 274 is the Stirling number of the first kind \(S(6,2).\) Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth Second smallest Euler brick, sides \(275, 252,240\), which has all integer face diagonals Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 276 is the smallest number that is triangular, hexagonal, untouchable and centered pentagonal. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 277 is the smallest prime with a multiplicative persistence of 4 Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin There exists a positive integer that cannot be written as a sum of \(n\) 8th powers for all \(n\le278\). Alex Li · 1 year, 7 months ago

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@Alex Li Every positive integer is the sum of at most 279 8th powers.

279 is the smallest number whose digit product is 7 times its digit sum. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth \({ 2 }^{ 3 }\cdot 5\cdot 7=280=10!!!\)

(!!! means triple factorial, i.e. product of every third number) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 281 BC is the year Seleucus I Nicator, the Satrap of Babylon and the founder of the Seleucid Dynasty, died. He was sorely missed. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin And in 281 A.D., Emperor Probus returned to Rome to celebrate his triumph over the Vandals and Usurpers.

(On a lesser note, 281 is a Sophie Gemain prime and is the sum of the first 14 primes.)

Unfortunately for Emperor Probus, 282 A.D. was not as successful a year. He traveled to Sirmium, (Serbia), where he attempted to employ his troops in peaceful engineering projects such as the draining of the swamps in Pannonia.

Probus was subsequently murdered by his discontented troops. All hail Emperor Marcus Aurelius Carus!

282 is a preparation of aspirin with 15 mg of codeine. Perhaps Probus should have given some to his troops to make them less grouchy.

282 is also the smallest multi-digit palindrome sandwiched between twin primes. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 283 cubic inch small block Chevy V-8 engine in Corvette Stingrays, need I say more? Oh, wait, Eisenhower was president, television was black and white, Bridget Bardot was hot, and beatniks were subversive. When you heard about the Beetles, you called pest control.

\(\frac { 1 }{ 2 } \left( 6!-5!-4!-3!-2!-1!-0! \right) =283\) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Another year, another emperor.... In 284 A.D., Emperor Numerian, (who had succeeded his father Carus, (who died tragically after an unfortunate encounter with a bolt of lightning), the previous year), while traveling in a closed litter through Asia Minor on the way home to Rome, (to which all roads lead), found himself suddenly afflicted with an inflammation of the eyes. Upon return and after opening his litter, his decaying corpse was found. All hail Emperor Diocletian, who promptly blamed Arrius Aper, (a rival for the throne), for Numerian's death and had him executed on the spot. Apparently Diocletian and Arrius weren't that amicable, unlike ...

\(284\) and \(220,\) which together form the smallest amicable pair. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth And today, we think politicians like Donald Trump are so outrageous. They have nothing on the Romans.

\( { 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 }+{ 5 }^{ 2 }+{ 6 }^{ 2 }+{ 7 }^{ 2 }+{ 8 }^{ 2 }+{ 9 }^{ 2 }=285\)

Also, 285 AD marks the end of unified Roman Empire. But you knew that. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Deep down, we're no different than the ancient Romans; the players may change but the song remains the same. Provide swords to the participants in the upcoming Republican debate and the field will be cut down to size in no time. (Dr. Carson may have an advantage in that he knows where the human body's 'kill spots' are located, as would Rick "Oops!" Perry by way of his military training. Everyone would go for The Donald once the inevitable melee broke out. (He could use his hair as a shield, though, so he might survive.) Fox's ratings would go through the roof, better than those of the finale of "The Howard Beale Show" in "Network".)

And yes, in 286 A.D. Diocletian divided the Empire into two, appointing Maximian as co-emperor and giving him control over the Western Empire. Apparently the two were an amicable pair, with Maximian's military brilliance complementing Diocletian's political acumen. This branching out is mirrored by the number 286, in that .....

286 is the number of rooted trees on 9 nodes. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth \(287=89+97+101=47 + 53 + 59 + 61 + 67=17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47\) Alex Li · 1 year, 7 months ago

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@Alex Li The Bagger 288, (Excavator 288), built by the German company Krupp, is a mobile strip mining machine, and when built in 1978 was the heaviest land vehicle in the world at 13,500 tonnes, superseding The Big Muskie, a coal mining dragline. In 1995 The Bagger 288 was superseded in size by The Bagger 293, which weighs in at 14,200 tonnes. Both Baggers are capable of moving 240,000 cubic metres of earth per day.

288 is also known as the Feist number, since \(288 = 1^{1} + 2^{2} + 3^{3} + 4^{4}.\) Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 288, aka "too gross", is a superfactorial \(4!\cdot 3!\cdot 2!\cdot 1!=288\)

"Baggers" sounds more British than German.

289 cubic inch small block Ford V-8 in Shelby Mustang GT350, black with gold stripes.

Also, 289 is the square of the sum of the first four prime numbers. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin Oh, goodness, 288 is an exceptionally Feist-y number. And that car .... are we talking 1965-1967? A lot of muscle and not a lot of them produced; it must have been fun when it wasn't in the shop. It would seem then that you are, or at least were, a car enthusiast; a love/hate endeavor if ever there was one.

As for our Roman friends, in 290, Diocletian and Maximian reluctantly acknowledge Carausius, who has established himself as king of Britain, as third Emperor. The more the merrier ..... We can also celebrate the birth of Pappus of Alexandria, the last great Greek mathematician of Antiquity, who gave us Pappus's Theorem, among other gems.

Pappus may not have known this, but 290 is the smallest sphenic, untouchable number that is also an element of the Mian-Chowla sequence. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth In my foolish youth, I bought a 1966 Shelby Mustang GT350 from a Vanderburg AF guy who had cooked the hood from overheating his engine. It was either that or get a Steinway piano, I didn't have money for both. For years afterwards, it was my ride during the time I worked in South Central LA--as the only whitey in town while driving a cooked Shelby Mustang GT350. Interesting times. Eventually, the engine blew up in freeway traffic, and I sold it cheap to collectors two days later. The problem with the car is that it had a lot of muscle, but the under-built Mustang was mechanically unable to deal with the boost--so it just kept breaking down, and I kept fixing it forever. It just wore me down, and the engine explosion was the last straw.

291 is the largest number that cannot be expressed by a nontrivial sum of powers of integers. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin In the movies, Air Force guys always seem to drive around in Mustangs, and apparently in real-life, too. The Steinway would have been nice, but you can't drive it to work. I just did a quick read on South Central LA; 1.2% white .... interesting times, indeed. Muscle cars almost seem like a rite of passage for young guys; my first car was a Custom 1970 Buick Skylark, with a 350 V8. The engine didn't blow up, but I eventually sold it to a collector after getting tired of spending most of my money on gas.

Speaking of money, 292 is the number of ways (using standard coin denominations) of making change for a dollar.

292 is also rather conspicuous in the continued fraction representation of \(\pi.\) Truncation at this term yields the approximation \(\frac{355}{113} = 3.1415929...,\) correct to the 6th decimal place. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth I decided to check up on South Central LA on Wikipedia. Here's a quote, which matches my experience there, "Nonetheless, South Los Angeles remains home to the largest black community in the Western United States." Yeah, I got to meet with a lot of them, all right. Those were the days when the area was red-lined off, ghettoized by bureaucratic fiat, and property values fell down to ridiculous lows, soon after the Watts Riots. I had trouble even buying gasoline for my car in there. Well, things are very different now, and I hope I've helped to make a difference.

293 is all kinds of primes. Let's see, besides being a regular prime, it's

1) Sophie Germain prime
2) Pythagorean prime
3) Single prime
4) Chen prime
5) Irregular prime
6) Eisenstein prime
7) Strictly nonpalindromic prime
8) Happy number prime
9) Right truncatable prime
10) and even a "Bad" prime Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin After all the depressing headlines of late it's good to be reminded that there has been some progress over the years, and I'm glad to hear that you played a part in that progress. There's still so much to be done, particularly with regard to rooting out police racism, and I hope that the recent spotlight on it will result in an accelerated improvement in the relationship between the police and minorities.

294 is the number of biconnected planar graphs with 8 nodes.

Also, \(11115^{2} - 294^{2} = 123,456,789.\) Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 295 is the only second Lychrel number. A Lychrel number is a number that never (or is not known to ever) become palindromic after repeated "add the number's reversal to itself", even after millions of such reiterations resulting in numbers millions of digits long. The latest efforts to produce a palindrome from the first such Lychrel number, 196, has produced a number nearly a billion digits long, and still not palindromic.

The problem of racism in America is a very complicated subject, and it's an universal human condition. To imagine that we're living in some kind of a post-racist society is just fantasy . Racism is not rooted in rational thought, and people generally aren't as rational as they could or should be. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin As long as there is difference, the "Other", there will be racism. It's unsettling to realize that we are all guilty of it to some degree and that it will always be with us, but awareness is the first step to at least lessen its negative impact.

I hadn't heard of Lychrel numbers before. Finding them seems like a rather odd pursuit, but the fact that there is presently no actual proof that any base-10 Lychrel numbers exist does make them at least a bit intriguing.

296 is the number of partitions of 30 into distinct parts. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth Brian Sir , when do you sleep? :P

297 is the smallest 3-digit Kaprekar number.

\[297^2=88209 \rightarrow 88+209=297\] Nihar Mahajan · 1 year, 7 months ago

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@Nihar Mahajan $2.98 is the most common price at the Dollar Store? Gee, 298 is boring. Have we finally found our candidate?

Okay, 298 squared is 88804, the only second number to have the property of its square starting with three identical digits. (Very Hard Problem--find the first!) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 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.)

Anyway, with 298 only being the second such number, I don't think it qualifies as being "special". I can't find anything else particularly interesting about it, (not even as an historical date), so I'll nominate it as the most boring number up to 300 and see if you want to second it. Meanwhile, back at the number farm ....

299 is the smallest (positive) number whose digit sum is 20.

299 is the maximum number of pieces into which a cake can be sliced with 12 plane cuts.

@Nihar Mahajan I sleep when I'm tired. :) I'm a night owl, for sure. Brian Charlesworth · 1 year, 7 months ago

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@Brian Charlesworth 300 is maximum score in bowling. Also

1) Spartans at the Battle of Thermoplyae (how many movies have been made about this already?)
2) Gideon's followers against the Midianites
3) Israeli king Thalut's solders against Goliath's men
4) Muhammad's followers in the Battle of Sadr
5) Jewish family followers of Sabbatai Zevi (later forced to convert to Islam)
6) Length of Noah's Ark in cubits
7) Foxes captured by Samson, which he set on fire and let through the crops of the Philistines
8) Disciples of Pythagoras

Moral of the story: If you want to get something done, get 300 of whatever is it that you need.

300 is the Pisano period of the two rightmost digits of Fibonacci numbers

300 is the largest known number that is not the sum of a prime and a number that has exactly three prime factors

Brian, maybe we should pause this list for the time being, as to give us the chance to identify and vote which of the 300 wins as the most boring number. You've nominated 298. Let me find more. (Meanwhile, we can let others continue). Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin That sounds like a good idea. 300 is quite special so is a good number to take a break at and evaluate what we have so far. I'll look back over the list as well to see if 298 is the nomination I want to stick with. Perhaps others following this thread might want to make nominations as well. Brian Charlesworth · 1 year, 7 months ago

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@Michael Mendrin \(301\) is the target score for a certain variant in darts.

I nominate \(284\) as the most boring number. Alex Li · 1 year, 7 months ago

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@Alex Li Here is the quest to find the most boring number. I have added Brian's vote of 298 and Alex's vote of 284. Calvin Lin Staff · 1 year, 7 months ago

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@Michael Mendrin Also, 281 is the most known number of primes that can be found in a 7x7 matrix, see Whiteside. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 82000 : least (perhaps the only) number that contains only 0's and 1's in bases 2, 3, 4 & 5. Milind Chakraborty · 1 year, 7 months ago

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@Milind Chakraborty Excellent! Now we need to examine integers from 301 to 81999. Do you want to start? Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin 301 = 7 × 43. 301 is the sum of three consecutive primes (97 + 101 + 103), happy number in base 10. Milind Chakraborty · 1 year, 7 months ago

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