# What is the smallest integer you can think of that is not interesting?

Asked by ETpro (34600) March 23rd, 2010

Most small integers are interesting for some reason. Zero is interesting because any number multiplied it = 0. 1 is interesting because any number multiplied by it is that number itself. 2 is interesting because it is the smallest even number. 3 is the smallest number that defines a 2-dimensional shape—the triangle. 4 is the first composite number. 5 is (among many interesting things) the number of regular polyhedra in three dimensions. 6 is very interesting because it is 3 factorial, (3 x 2 x 1) and also is (3 + 2 + 1). You get the drift. Most small integers are interesting for one or more reasons.

But surely at some point, we hit an integer that isn’t a square of another, isn’t interesting in any particular way. Now that in itself develops an interesting paradox, because such a number is VERY interesting in that it is the smallest number that isn’t interesting for any other reason. But setting that paradox aside, what number would you say deserves the title of smallest uninteresting integer?

For hints, you could consult The Penguin Dictionary of Curious and Interesting Numbers by David Wells and Les Nombres Remarquables by François Le Lionnais. But surely someone here can point out why their conclusions are off base, and the lowest integer they each site is actually quite interesting on its own merits, and thus doesn’t need the cover of the aforementioned paradox.

Asking here is the best way I know to shoot down small numbers one by one till we reach some number none of us find interesting except in that it does fit our “Interesting for being uninteresting” paradox.

Observing members: 0 Composing members: 0

I think it’s 3. All interest ends with the awesome 2 – the only even prime number.

hug_of_war (10735)

I think 1 is pretty boring.

Lightlyseared (34387)

I think seven. It’s prime, but it’s not the smallest prime, or even second-smallest.

hannahsugs (3243)

1 isn’t fun.

erichw1504 (26443)

if you consider 7 interesxsting, it is probably not for a mathematical reason.
7

anartist (14803)

Yeah, my vote is for 7 also.

Although if we’re really going to stretch it, seven is the only one digit number whose inverse is not terminating or a single number repeated… although that’s not very interesting.

Mariah (25883)

@Mariah Thats not really that interesting though is it.

Lightlyseared (34387)

Let’s say any prime number counts as interesting just because it’s prime. Then seven doesn’t count anymore. If we continue on, eight is a perfect cube, nine is a perfect square, 10 is the base of our counting system. eleven is prime, twelve is one dozen, and is divisible by 2, 3, 4, and 6. Thirteen is prime, fourteen is…..boring? Can any one argue any particularly interesting things about 14?

hannahsugs (3243)

I’ll vote for 14 as well.

cockswain (15276)

@cockswain Integers don’t include decimals; they are numbers like 0, 1, 2, 3…

Although, he just reminded me that integers DO include negative numbers (-1, -2, -3…) so if we truly want to find the smallest uninteresting integer, we should be looking to the negatives right now. I think we want to be looking at whole numbers, which start at 0 (0, 1, 2, 3…). Just a picky thing. :]

EDIT: Oh, Ninja’d, whoops. Sorry @cockswain.

Mariah (25883)

s’okay, I noticed my mistake as soon as I posted. Otherwise someone else would have obviously said 0.02.

cockswain (15276)
timtrueman (5765)

I believe that settles it actually.

cockswain (15276)

391 is the first integer that is both centered pentagonal and a smith number.

Sorry.

nisse (1986)

Haha, nice work. You should email the guy who maintains that page…

timtrueman (5765)

Thanks for all who took a crack at it. Special shout out and GA to @hannahsugs for debunking 7, to @timtrueman for taking an unscientific WAG and praticularly to @nisse for using a scientific approach to debunk 391.

I’ll give you this headstart. Both the two books mentioned in the question have different guesses, but the smallest number both agree is uninteresting is 62, so you might want to start by proving they are both wrong about that. :-)

ETpro (34600)

62 is the smallest number that can be written as the sum of of 3 distinct squares in 2 ways. That’s on the list provided by @timtrueman.

finkelitis (1907)

11011400010 is the smallest positive integer i can find that doesn’t yield any hits on Google.

nisse (1986)

Personal I would say all numbers are interesting—anyways—why would you say my answer is an unscientific wild-ass guess?

timtrueman (5765)

@finkelitis Excellent debunking. I didn’t think that would take long.

@nisse Please tell me you programmed a bot to do the work on that. In any case, That’s Interesting. But soon, this post will have it too on Google. :)

@timtrueman I made a unscientific WAG. In what fashion did you arrive at 391?

ETpro (34600)

@ETpro check out the link @timtrueman posted. Just click on 391 in his post.

finkelitis (1907)

@ETpro: thanks for the props, but i also proposed seven in the first place, then changed my mind ;)

hannahsugs (3243)

@finkelitis & @timtrueman Thanks for calling my attention to that, and my apologies. We should call the list authro’s attention to what’s interesting about 391. :-)

In case no-one else has, I just emailed the maintainer of that page and called his attention to this thread.

ETpro (34600)

@ETpro: No, that number was only found using some rudimentary trial and error. No worries about the paradox though, i’ve since found quite a few zero-hit integers that are smaller than the one i gave, but i won’t name any here (both as a challenge, as well as not to ruin the paradox. ;).

However your question inspired me so i wrote a bot that googled the numbers 0–1000. It turns out that according to Google, the five most uninteresting integers below 1000 are:
983 – 982 – 977 – 976 – 959 (in order of increasing popularity).

Here’s a chart of the popularity: http://i.imgur.com/rW51U.png

Some perhaps interesting observations:

- There are two unexpected sharp drops in popularity, 31–32 is the first, 60–61 the second.

- The ten most popular integers below 1000 are
1 – 2 – 3 – 10 – 4 – 5 – 0 – 12 – 6 – 11 (in order of decreasing popularity).
This verifies what previous posters have conluded with “7” not being a very interesting number (although to be fair “7” is number 11 on the popularity list).

- As can be seen in the chart, lower numbers are more popular (perhaps not an unexpected conclusion). To see if i could debias this unhealthy obsession with small numbers, I computed debiased popularity as = N*Number of hits for number(N).

With debiased popularity the least popular number is not unexpectedly “0” (with a debiased popularity rating of 0), followed by
1 – 2 – 3 – 394 (!) (in order of increasing popularity).

This shows that 394 is actually a very uninteresting integer compared to its relatively small size.

nisse (1986)

@nisse Wonderful work. 394 is a US Interstate Highway as well as a popular prefix in US phone numbers for a host of cities and a telephone area code in Jamaica. So along with its other interesting properties, this gets it a good deal of attention in the Google bot’s eyes. :-)

ETpro (34600)

This is a strong contender for my favorite thread ever.

nikipedia (28071)

@nikipedia Why thank you. With my respect for your intelligent comments and questions, and the length of time you have been here, that means a great deal to me.

ETpro (34600)

@ETpro. Actually, 394 was my contender for most un-interesting small integer (not most interesting). :)

Perhaps my description of that last conclusion was bad, what i did was weight the number of hits by the size of the number. Hence the number of hits for “1” was weighted (multiplied) by 1, number of hits for 2 was weighted by 2, number of hits for 1000 weighted by 1000. Doing it that way it turned out that 394 had the fifth lowest score.

I would like to add that I (in opposition to the crowd) find the number 7 absolutely fascinating. Just look at the properties of 1/7=0.142857…

1 * 142 857 = 142 857
2 * 142 857 = 285 714 (same numbers, shifted!)
3 * 142 857 = 428 571 (again, shifted)
4 * 142 857 = 571 428 (...)
5 * 142 857 = 714 285
6 * 142 857 = 857 142
7 * 142 857 = 999 999 (why does it stop shifting the numbers all of a sudden?)

Also,
8 * 142 857 = 1 142 856, add the first digit to the last digit, what do you get?
142 857 * 142 857 = 20 408 122 449, add the first five digits to the last five, what do you get? (hint: 142 857)

142 + 857 = 999
428 + 571 = 999
285 + 714 = 999
14 + 28 + 57 = 99
42 + 85 + 71 = 99 + 99
1 + 4 + 2 + 8 + 5 + 7 = 9 + 9 + 9

nisse (1986)
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