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

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

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

**Rules:**

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

2. Make sure you use 14 words or less.

3. Do not reply out of sequence.

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

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

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

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

which is a contradiction.

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## Comments

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

Reply to this thread :)

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

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

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

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This is in reply to what number?

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Brian's comment

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

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

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

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

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

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

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

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

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

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

eh

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

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

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

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

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

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

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

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

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

Base 6, 42, 85, 171.

Single digit numbers are not considered repdigits.

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

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

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

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

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

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

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Comment deleted Jul 24, 2015

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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

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185 is the number of conjugacy classes in the automorphism group of the 8-dimensional hypercube.

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Meanwhile, time for me to go to sleep.

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Every Integer..., the \(19th\) of this month (July). See \(19\) in this list. Was gone the whole weekend to have fun.

As noted inLog in to reply

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

(GF(2) is a Galois field of 2 elements.)

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Also 187 is the California Penal Code for murder.

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semigroups of order 4.

188 is the number of nonisomorphicLog in to reply

E.g.

\(821\cdot 823\cdot 827\cdot 829=463236778189\)

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(A Roman palindrome is an integer that is palindromic when written in Roman numerals.)

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The 191 orientable...

Also, $1.00 + $0.50 + $0.25 + $0.10 + $0.05 + $0.01 in US coins = $1.91

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192 is the smallest number with 14 divisors.

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Keith number.

197 is the only prime 3-digitLog in to reply

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Lucas number that is also a permutable prime.

199 is the only 3-digitLog in to reply

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102 Brew

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

Otherwise, 201 is the smallest pentadecagonal and icosagonal number.

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\(202 = (2 + 3 + 5 + 7)^{2} - (2^{2} + 3^{2} + 5^{2} + 7^{2}).\)

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\(2+5+7+43+61+89=207\)

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necklaces using beads of 4 colors.

208 is the number of 5-beadLog in to reply

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primorial).

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

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\(1\cdot 8\cdot 9+2\cdot 5\cdot 7+3\cdot 4\cdot 6=214\)

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\(\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} + \dfrac{1}{d} = 1.\)

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

\({ 3 }^{ 3 }+{ 4 }^{ 3 }+{ 5 }^{ 3 }={ 6 }^{ 3 }=216\)

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You know, Brian, I think these number facts offer possibilities as Brilliant.org problems.

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Seems like a somewhat painful counting / case checking though.

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219 is the smallest number expressible as a sum of four positive cubes in two ways.

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220 is the largest difference between two consecutive primes for all primes < 100,000,000

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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.

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prime island and the least prime whose adjacent primes differ by 16.

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 \(p+q=r\)

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225 is the smallest square to have one of every digit in some base (3201 in base 4).

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One US gallon is exactly 231 cubic inches.

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

232 is the number of symmetric permutation matrices.

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Brian, I have no idea why the notable Mendrin-Hardy-Har numbers is not listed in OEIS.

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234 is a postage stamp problem solution for 4 denominations and 8 stamps.

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

\(40, 45, 175, 224, 1573, 5491, 26071, 26624, 72283,...\)

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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. (\(2*3*6 = 36, 2 + 2 + 59 = 63.\))

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237 is the smallest number such that the first three multiples of it has the digit 7

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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.)

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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.

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Also, \(n^{m} - n^{m-4}\) is divisible by 240 for any integer \(m \gt 7, n \in \mathbb{Z^{+} \cup \{0\}}.\)

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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?

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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.

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243 is the largest 3-digit number that is a fifth power \((3^5)\).

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(left) concatenation of all numbers \(\le n\) written in base 2.

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

\(246=2\cdot 3\cdot 41\)

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(\(247 = 50123 - 49876.)\)

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Sure major Texan highway, it goes all the way to Tomball and beyond out from Houston.

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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).

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In other words, I nominate 249 as the most boring number up to 250. Any seconders?

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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".

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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.

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star number.

253 is the smallest non-trivial triangularLog in to reply

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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...

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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.

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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 \(k.\)

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order 64.

267 is the number of groups ofLog in to reply

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Owls are able to rotate their heads a full \(270^{\circ}.\)

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Stirling number of the first kind \(S(6,2).\)

274 is theLog in to reply

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untouchable and centered pentagonal.

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

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279 is the smallest number whose digit product is 7 times its digit sum.

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(!!! means triple factorial, i.e. product of every third number)

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(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.

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\(\frac { 1 }{ 2 } \left( 6!-5!-4!-3!-2!-1!-0! \right) =283\)

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\(284\) and \(220,\) which together form the smallest amicable pair.

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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.

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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.

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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 \(288 = 1^{1} + 2^{2} + 3^{3} + 4^{4}.\)

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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.

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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.

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291 is the largest number that cannot be expressed by a nontrivial sum of powers of integers.

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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.

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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

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

Also, \(11115^{2} - 294^{2} = 123,456,789.\)

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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.

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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.

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297 is the smallest 3-digit Kaprekar number.

\[297^2=88209 \rightarrow 88+209=297\]

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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!)

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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.

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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).

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I nominate \(284\) as the most boring number.

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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.

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happy number in base 10.

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