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.

## Comments

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

Reply to this thread :) – Nihar Mahajan · 1 year, 4 months ago

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– Brian Charlesworth · 1 year, 4 months ago

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

– Michael Mendrin · 1 year, 4 months ago

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

@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, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

This is in reply to what number?Log in to reply

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, 10 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, 10 months ago

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– Michael Mendrin · 1 year, 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.Log in to reply

– Calvin Lin Staff · 1 year, 10 months ago

Yup! That's one thing I'm looking for.Log in to reply

\({ 1 }^{ 3 }+7^{ 3 }+{ 3 }^{ 3 }=371\) – Michael Mendrin · 1 year, 10 months ago

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That's a pretty interesting fact to come across. Only prime? Hm.... – Calvin Lin Staff · 1 year, 10 months ago

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

eh – Michael Mendrin · 1 year, 10 months ago

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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 \( a_i, b_i , c_i \). – Calvin Lin Staff · 1 year, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

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.Log in to reply

The theorem Calvin mentioned seems like a considerably more formidable one to prove. – Michael Mendrin · 1 year, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

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.Log in to reply

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, 10 months ago

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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 \( c > 1 \) ). – Calvin Lin Staff · 1 year, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

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.Log in to reply

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, 10 months ago

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@Calvin Lin What are the guidelines for the length of comment, especially if it contains numbers or symbols? – Michael Mendrin · 1 year, 10 months ago

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Base 6, 42, 85, 171.

Single digit numbers are not considered repdigits. – Calvin Lin Staff · 1 year, 10 months ago

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– Alex Li · 1 year, 10 months ago

Element 173 is thought to be the highest possible element.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

Also, 173 is a Cuban prime, i.e., a prime number which is the difference between two consecutive cubes.Log in to reply

(The bracketed phrase can be deleted if necessary to comply with the 14-word limit.) – Brian Charlesworth · 1 year, 10 months ago

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@Michael Mendrin We have a couple of parallel threads going here, so I figure we should streamline on this one. – Brian Charlesworth · 1 year, 10 months ago

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Let's see if you can do it without mentioning the Declaration of Independence. – Michael Mendrin · 1 year, 10 months ago

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176 is the number of possible partitions of the number 15. – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

177 is the smallest magic constant for a 3x3 prime magic squareLog in to reply

– Nihar Mahajan · 1 year, 10 months ago

178 is a digitally balanced number, as its binary number (10110010) has an equal number of zeros and ones.Log in to reply

\({ 179 }^{ 2 }=32041\) – Michael Mendrin · 1 year, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

180 is the sum of the interior angles of a triangle in Euclidean space.Log in to reply

(Strobogrammatic means it looks the same reversed left-right, or up-down) – Michael Mendrin · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

182 is the first pronic sphenic number not divisible by 10Log in to reply

– Brian Charlesworth · 1 year, 10 months ago

183 is the smallest \(n\) such that \(n\) concatenated with \(n + 1\) is a square.Log in to reply

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, 10 months ago

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185 is the number of conjugacy classes in the automorphism group of the 8-dimensional hypercube. – Brian Charlesworth · 1 year, 10 months ago

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Meanwhile, time for me to go to sleep. – Michael Mendrin · 1 year, 10 months ago

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– Nihar Mahajan · 1 year, 10 months ago

I see that you have turned 65. When was your birthday?Log in to reply

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, 10 months ago

As noted inLog in to reply

– Nihar Mahajan · 1 year, 10 months ago

Oh , I see.Log in to reply

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, 10 months ago

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Also 187 is the California Penal Code for murder. – Michael Mendrin · 1 year, 10 months ago

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semigroups of order 4. – Brian Charlesworth · 1 year, 10 months ago

188 is the number of nonisomorphicLog in to reply

E.g.

\(821\cdot 823\cdot 827\cdot 829=463236778189\) – Michael Mendrin · 1 year, 10 months ago

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(A Roman palindrome is an integer that is palindromic when written in Roman numerals.) – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

Also, there are only 190 polycubes of order 4. No boring number nomination for this one.Log in to reply

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, 10 months ago

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192 is the smallest number with 14 divisors. – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

193, when divided by 71, gives the best approximation of e using 5 digits or fewerLog in to reply

– Alex Li · 1 year, 10 months ago

The sum of the proper divisors of 194 is 100.Log in to reply

– Alex Li · 1 year, 10 months ago

There are 195 sovereign countries in the world.Log in to reply

– Alex Li · 1 year, 10 months ago

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.Log in to reply

Keith number. – Brian Charlesworth · 1 year, 10 months ago

197 is the only prime 3-digitLog in to reply

– Michael Mendrin · 1 year, 10 months ago

Also Circular Prime and Super Catalan. No boring number here.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

198 is one of the most commonly posted prices in storesLog in to reply

– Michael Mendrin · 1 year, 10 months ago

Also Pell-Lucas number. Not boring.Log in to reply

Lucas number that is also a permutable prime. – Brian Charlesworth · 1 year, 10 months ago

199 is the only 3-digitLog in to reply

– Michael Mendrin · 1 year, 10 months ago

200 is the smallest number which cannot be changed into a prime by changing one digit (orly!)Log in to reply

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, 10 months ago

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\(202 = (2 + 3 + 5 + 7)^{2} - (2^{2} + 3^{2} + 5^{2} + 7^{2}).\) – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

The product of the divisors \( 1\cdot 7\cdot 29=203 \), where \(1729\) is the Hardy-Ramanujan NumberLog in to reply

– Brian Charlesworth · 1 year, 10 months ago

204 is the number of different squares on a chessboard.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

There are 205 twin primes less than 10,000Log in to reply

– Michael Mendrin · 1 year, 10 months ago

Also both Primitive Sequence number and Wolstenholme number. No nomination here.Log in to reply

– Brian Charlesworth · 1 year, 10 months ago

206 is the number of bones in a typical adult human body.Log in to reply

\(2+5+7+43+61+89=207\) – Michael Mendrin · 1 year, 10 months ago

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necklaces using beads of 4 colors. – Brian Charlesworth · 1 year, 10 months ago

208 is the number of 5-beadLog in to reply

– Michael Mendrin · 1 year, 10 months ago

209 is the smallest that can be expressed as a sum of three squares in six different waysLog in to reply

primorial). – Brian Charlesworth · 1 year, 10 months ago

210 is the product of the first 4 primes, (and hence the only 3-digitLog in to reply

– Michael Mendrin · 1 year, 10 months ago

211 is the largest known prime number that is not a sum of a prime and a triangular numberLog in to reply

– Brian Charlesworth · 1 year, 10 months ago

\(212^{\circ}\) Fahrenheit is the boiling point of water at sea level.Log in to reply

– Julian Poon · 1 year, 10 months ago

\(213^{2}=1!+2!+3!+7!+8!\)Log in to reply

\(1\cdot 8\cdot 9+2\cdot 5\cdot 7+3\cdot 4\cdot 6=214\) – Michael Mendrin · 1 year, 10 months ago

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\(\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} + \dfrac{1}{d} = 1.\) – Brian Charlesworth · 1 year, 10 months ago

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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, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

217 is the sum of the positive divisors of 100, i.e., \(\sigma_{1}(100) = 217.\)Log in to reply

You know, Brian, I think these number facts offer possibilities as Brilliant.org problems. – Michael Mendrin · 1 year, 10 months ago

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Seems like a somewhat painful counting / case checking though. – Calvin Lin Staff · 1 year, 10 months ago

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219 is the smallest number expressible as a sum of four positive cubes in two ways. – Brian Charlesworth · 1 year, 10 months ago

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220 is the largest difference between two consecutive primes for all primes < 100,000,000 – Michael Mendrin · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

Also, only the 4th smallest power of 2 that contains the digits 666. Too cool to be boring.Log in to reply

prime gap table in mind for future numbers. This next one is a personal favourite...

Huh, I'll have to keep the221(B) Baker Street, London, England, is the (fictional) address of Sherlock Holmes. – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

222 is smallest three digit number with prime only digits (!)Log in to reply

prime island and the least prime whose adjacent primes differ by 16. – Brian Charlesworth · 1 year, 10 months ago

223 is aLog in to reply

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

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, 10 months ago

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225 is the smallest square to have one of every digit in some base (3201 in base 4). – Pranshu Gaba · 1 year, 10 months ago

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– Nihar Mahajan · 1 year, 10 months ago

At most 226 different permutation patterns can occur within a single 9-element permutation.Log in to reply

– Pranshu Gaba · 1 year, 10 months ago

The 227th harmonic number is the first to exceed six.Log in to reply

– Nihar Mahajan · 1 year, 10 months ago

There are 228 matchings in a ladder graph with five rungsLog in to reply

– Pranshu Gaba · 1 year, 10 months ago

229: Smallest prime, when added to the reverse of its decimal representation, yields another prime (229 + 922 = 1151)Log in to reply

– Nihar Mahajan · 1 year, 10 months ago

There are 230 unique space groups describing all possible crystal symmetries.Log in to reply

One US gallon is exactly 231 cubic inches. – Michael Mendrin · 1 year, 10 months ago

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

232 is the number of symmetric permutation matrices. – Brian Charlesworth · 1 year, 10 months ago

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Brian, I have no idea why the notable Mendrin-Hardy-Har numbers is not listed in OEIS. – Michael Mendrin · 1 year, 10 months ago

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234 is a postage stamp problem solution for 4 denominations and 8 stamps. – Brian Charlesworth · 1 year, 10 months ago

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

\(40, 45, 175, 224, 1573, 5491, 26071, 26624, 72283,...\) – Michael Mendrin · 1 year, 10 months ago

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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, 10 months ago

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237 is the smallest number such that the first three multiples of it has the digit 7 – Michael Mendrin · 1 year, 10 months ago

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untouchable number that is the sum of the first \(n\) primes for any \(n \gt 2.\)

238 is the smallest(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, 10 months ago

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

Also, 239 is the largest number that cannot be the sum of 8 cubes or less. – Michael Mendrin · 1 year, 10 months ago

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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, 10 months ago

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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, 10 months ago

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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, 10 months ago

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243 is the largest 3-digit number that is a fifth power \((3^5)\). – Nihar Mahajan · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

244 is the smallest non-trivial number that is both the sum of two squares and the sum of two fifth powers.Log in to reply

(left) concatenation of all numbers \(\le n\) written in base 2. – Brian Charlesworth · 1 year, 10 months ago

245 is the greatest 3-digit number \(n\) that divides theLog in to reply

\(246=2\cdot 3\cdot 41\) – Michael Mendrin · 1 year, 10 months ago

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(\(247 = 50123 - 49876.)\) – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

248 is the smallest hexanacci number with digits in geometric progressionLog in to reply

– Alex Li · 1 year, 10 months ago

249 is a major highway in Texas.Log in to reply

Sure major Texan highway, it goes all the way to Tomball and beyond out from Houston. – Michael Mendrin · 1 year, 10 months ago

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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, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

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!Log in to reply

In other words, I nominate 249 as the most boring number up to 250. Any seconders? – Brian Charlesworth · 1 year, 10 months ago

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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, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

Well, o.k., I suppose I'll let 249 scrape by, then. :)Log in to reply

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, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

\(\dfrac { 10\cdot 9\cdot 8\cdot 7\cdot 6 }{ 5\cdot 4\cdot 3\cdot 2\cdot 1 } =252\)Log in to reply

star number. – Brian Charlesworth · 1 year, 10 months ago

253 is the smallest non-trivial triangularLog in to reply

– Michael Mendrin · 1 year, 10 months ago

254 is the first nonzero number of the form \({ n }^{ 8 }-n!\)Log in to reply

– Brian Charlesworth · 1 year, 10 months ago

255 is the greatest integer that can be represented as an 8-digit binary number.Log in to reply

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, 10 months ago

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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, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

258 is the smallest magic constant for a 4x4 magic square using 16 consecutive primes.Log in to reply

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, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

260 is the magic constant in a 8x8 magic square using numbers 1 through 64Log in to reply

– Brian Charlesworth · 1 year, 10 months ago

261 is the number of different ways to dissect a hexadecagon, (16-gon), into 7 quadrilaterals.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

262 different polykites can be formed from 7 kitesLog in to reply

– Nihar Mahajan · 1 year, 10 months ago

263 is an irregular prime , since it divides the numerator of the Bernoulli number \(B_{100}\).Log in to reply

– Michael Mendrin · 1 year, 10 months ago

264 is the largest known number which square consists of alternating digits 69696Log in to reply

– Nihar Mahajan · 1 year, 10 months ago

265 is the number of derangements of 6 elements.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

\({ 2 }^{ { 2 }^{ 3 } }+{ 2 }^{ 3 }+2=266\)Log in to reply

order 64. – Brian Charlesworth · 1 year, 10 months ago

267 is the number of groups ofLog in to reply

– Michael Mendrin · 1 year, 10 months ago

268 is the smallest number which product of its digits is six times the sum of its digitsLog in to reply

– Nihar Mahajan · 1 year, 10 months ago

269 is the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix.Log in to reply

Owls are able to rotate their heads a full \(270^{\circ}.\) – Brian Charlesworth · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

271 is the first three digits of Euler's constantLog in to reply

– Alex Li · 1 year, 10 months ago

272 is a Euler zigzag number.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

-273 is Kelvin 0Log in to reply

Stirling number of the first kind \(S(6,2).\) – Brian Charlesworth · 1 year, 10 months ago

274 is theLog in to reply

– Michael Mendrin · 1 year, 10 months ago

Second smallest Euler brick, sides \(275, 252,240\), which has all integer face diagonalsLog in to reply

untouchable and centered pentagonal. – Brian Charlesworth · 1 year, 10 months ago

276 is the smallest number that is triangular, hexagonal,Log in to reply

– Michael Mendrin · 1 year, 10 months ago

277 is the smallest prime with a multiplicative persistence of 4Log in to reply

– Alex Li · 1 year, 10 months ago

There exists a positive integer that cannot be written as a sum of \(n\) 8th powers for all \(n\le278\).Log in to reply

279 is the smallest number whose digit product is 7 times its digit sum. – Brian Charlesworth · 1 year, 10 months ago

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(!!! means triple factorial, i.e. product of every third number) – Michael Mendrin · 1 year, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

281 BC is the year Seleucus I Nicator, the Satrap of Babylon and the founder of the Seleucid Dynasty, died. He was sorely missed.Log in to reply

(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, 10 months ago

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\(\frac { 1 }{ 2 } \left( 6!-5!-4!-3!-2!-1!-0! \right) =283\) – Michael Mendrin · 1 year, 10 months ago

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\(284\) and \(220,\) which together form the smallest amicable pair. – Brian Charlesworth · 1 year, 10 months ago

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

Also, 285 AD marks the end of unified Roman Empire. But you knew that. – Michael Mendrin · 1 year, 10 months ago

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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, 10 months ago

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– Alex Li · 1 year, 10 months ago

\(287=89+97+101=47 + 53 + 59 + 61 + 67=17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47\)Log in to reply

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.

The288 is also known as the Feist number, since \(288 = 1^{1} + 2^{2} + 3^{3} + 4^{4}.\) – Brian Charlesworth · 1 year, 10 months ago

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"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, 10 months ago

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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, 10 months ago

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291 is the largest number that cannot be expressed by a nontrivial sum of powers of integers. – Michael Mendrin · 1 year, 10 months ago

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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, 10 months ago

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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, 10 months ago

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

Also, \(11115^{2} - 294^{2} = 123,456,789.\) – Brian Charlesworth · 1 year, 10 months ago

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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, 10 months ago

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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, 10 months ago

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297 is the smallest 3-digit Kaprekar number.

\[297^2=88209 \rightarrow 88+209=297\] – Nihar Mahajan · 1 year, 10 months ago

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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, 10 months ago

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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, 10 months ago

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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, 10 months ago

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– Brian Charlesworth · 1 year, 10 months ago

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.Log in to reply

I nominate \(284\) as the most boring number. – Alex Li · 1 year, 10 months ago

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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, 10 months ago

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– Michael Mendrin · 1 year, 10 months ago

Also, 281 is the most known number of primes that can be found in a 7x7 matrix, see Whiteside.Log in to reply

– Milind Chakraborty · 1 year, 10 months ago

82000 : least (perhaps the only) number that contains only 0's and 1's in bases 2, 3, 4 & 5.Log in to reply

– Michael Mendrin · 1 year, 10 months ago

Excellent! Now we need to examine integers from 301 to 81999. Do you want to start?Log in to reply

happy number in base 10. – Milind Chakraborty · 1 year, 10 months ago

301 = 7 × 43. 301 is the sum of three consecutive primes (97 + 101 + 103),Log in to reply