Interesting Numbers (Whole Numbers)

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Girl-With-A-Math-Fetish
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Interesting Numbers (Whole Numbers)

Postby Girl-With-A-Math-Fetish » Tue Dec 30, 2014 5:13 am UTC

What's the most interesting whole number in your opinion, and why? What makes that number interesting? I know Numberphile once made a video on "boring numbers," but I'm talking about what YOU think makes a number interesting.

I find primes interesting, and of course, twin primes. Numbers that are equivalent to xⁿ are rather interesting too (I particularly like 2ⁿ numbers). Fibonacci numbers are also interesting, especially when it comes to the prime factorization of its elements.


Also, speaking of primes, say that we can only test if a number n is prime by actually testing each number from 2 to ((n/2) - .5) out. What ways would we be able to shorten our testing process? Obviously if the number is even, we can immediately throw it out as composite. If the number ends in 5, we can also throw it out, as it'd be divisible by 5. If n's digits sum up to 9, we can throw it out, too. What other ways would we be able to reduce the number of tests we'd need to run?

There are many other things that make a number interesting (I think perfect numbers are interesting) in my point of view, but what do you think?
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Re: Interesting Numbers (Whole Numbers)

Postby somehow » Tue Dec 30, 2014 5:24 am UTC

In primality testing, you actually only need to test up to the square root of n. Also, if an integer's digits sum to something divisible by 3, the original integer must be divisible by 3 and thus composite (unless the integer is 3).
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Re: Interesting Numbers (Whole Numbers)

Postby Girl-With-A-Math-Fetish » Tue Dec 30, 2014 5:49 am UTC

somehow wrote:In primality testing, you actually only need to test up to the square root of n. Also, if an integer's digits sum to something divisible by 3, the original integer must be divisible by 3 and thus composite (unless the integer is 3).


But that wouldn't always work (the square root thing). It would only work in some circumstances. It would work for a number like 49, that isn't otherwise obviously composite. But we know that 49's square root is 7, so it's composite. But what of a number like 77? Both of us know that it's obviously divisible by 7 and 11. However it doesn't follow any of these other rules. And √77 is not a whole number. The test would only work in some circumstances.
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Re: Interesting Numbers (Whole Numbers)

Postby FancyHat » Tue Dec 30, 2014 6:10 am UTC

Girl-With-A-Math-Fetish wrote:But that wouldn't always work (the square root thing). It would only work in some circumstances. It would work for a number like 49, that isn't otherwise obviously composite. But we know that 49's square root is 7, so it's composite. But what of a number like 77? Both of us know that it's obviously divisible by 7 and 11. However it doesn't follow any of these other rules. And √77 is not a whole number. The test would only work in some circumstances.

It doesn't matter that 77 isn't square. Just get the square root, and check only those primes less than it to see if they're factors of it.

√77 = 8.77496...

If 77 is composite, then we can say 77=ab, where a and b are any positive integers that satisfy that equation. Now, if a>√77, b<√77. Otherwise, a<√77. So either way, we only need to check 77 for factors no greater than √77. 2? No. 3? No. 5? No. 7? Yes! 77/7=11.

Edited to add the following, because I forgot at first.

As for your original question of what I think is the most interesting whole number, and why, well, I don't know. Either it's relatively small, except all whole numbers are relatively small, or there isn't a most interesting whole number.

What I mean by all whole numbers being 'relatively small' is that each whole number is bigger than only a finite number of smaller whole numbers, while there are infinitely many bigger ones (obviously). So, no matter how huge a whole number is, it's still relatively small compared to most whole numbers. That's something I think is interesting itself.

And the reason I think there might not be a most interesting whole number is that I'm open to the possibility of ever-increasingly interesting whole numbers, a bit like how there are infinitely many primes. If there's a whole number that seems to be the most interesting because of some bunch of reasons, there might yet be an even larger whole number that's interesting for some more interesting bunch of reasons.

But for this, 'interesting' needs to be properly defined, and I just haven't done that. So I suppose I really just mean that I haven't sufficiently thought about what makes numbers 'interesting'.
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Re: Interesting Numbers (Whole Numbers)

Postby Girl-With-A-Math-Fetish » Tue Dec 30, 2014 8:32 am UTC

FancyHat wrote:
Girl-With-A-Math-Fetish wrote:But that wouldn't always work (the square root thing). It would only work in some circumstances. It would work for a number like 49, that isn't otherwise obviously composite. But we know that 49's square root is 7, so it's composite. But what of a number like 77? Both of us know that it's obviously divisible by 7 and 11. However it doesn't follow any of these other rules. And √77 is not a whole number. The test would only work in some circumstances.

It doesn't matter that 77 isn't square. Just get the square root, and check only those primes less than it to see if they're factors of it.

√77 = 8.77496...

If 77 is composite, then we can say 77=ab, where a and b are any positive integers that satisfy that equation. Now, if a>√77, b<√77. Otherwise, a<√77. So either way, we only need to check 77 for factors no greater than √77. 2? No. 3? No. 5? No. 7? Yes! 77/7=11.

Edited to add the following, because I forgot at first.

As for your original question of what I think is the most interesting whole number, and why, well, I don't know. Either it's relatively small, except all whole numbers are relatively small, or there isn't a most interesting whole number.

What I mean by all whole numbers being 'relatively small' is that each whole number is bigger than only a finite number of smaller whole numbers, while there are infinitely many bigger ones (obviously). So, no matter how huge a whole number is, it's still relatively small compared to most whole numbers. That's something I think is interesting itself.

And the reason I think there might not be a most interesting whole number is that I'm open to the possibility of ever-increasingly interesting whole numbers, a bit like how there are infinitely many primes. If there's a whole number that seems to be the most interesting because of some bunch of reasons, there might yet be an even larger whole number that's interesting for some more interesting bunch of reasons.

But for this, 'interesting' needs to be properly defined, and I just haven't done that. So I suppose I really just mean that I haven't sufficiently thought about what makes numbers 'interesting'.



Ahhhh I see. :3

And hehehe yep, that's why it's subjectively defined.
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Re: Interesting Numbers (Whole Numbers)

Postby Qaanol » Tue Dec 30, 2014 11:26 am UTC

Girl-With-A-Math-Fetish wrote:What's the most interesting whole number in your opinion, and why? What makes that number interesting?

You are aware of the interesting number paradox, yes?

Girl-With-A-Math-Fetish wrote:Also, speaking of primes, say that we can only test if a number n is prime by actually testing each number from 2 to ((n/2) - .5) out. What ways would we be able to shorten our testing process? Obviously if the number is even, we can immediately throw it out as composite. If the number ends in 5, we can also throw it out, as it'd be divisible by 5. If n's digits sum up to 9, we can throw it out, too. What other ways would we be able to reduce the number of tests we'd need to run?

Well, we could take advantage of the fact that PRIMES is in P for starters. Maybe prove the Riemann hypothesis after lunch as a follow-up?
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Re: Interesting Numbers (Whole Numbers)

Postby Girl-With-A-Math-Fetish » Tue Dec 30, 2014 1:16 pm UTC

Qaanol wrote:You are aware of the interesting number paradox, yes?

Well, we could take advantage of the fact that PRIMES is in P for starters. Maybe prove the Riemann hypothesis after lunch as a follow-up?


Now I am. x'D

Also, hehe, I had "Prove Navier-Stokes" booked for that time slot. Right after, though! I'll squeeze it in right before "Prove Collatz Conjecture."
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Re: Interesting Numbers (Whole Numbers)

Postby Qaanol » Tue Dec 30, 2014 7:35 pm UTC

As for interesting numbers, there are quite a few contenders.

0 and 1 are of course indispensable, but rather blasé.

-1, 2, e, i, φ, and π (or perhaps τ) occupy the next tier or two.

I might lean toward 6 personally (smallest product of distinct primes, largest order of a dihedral group that is also symmetric, smallest characteristic with a nontrivial ring but no fields, largest number of edges for a regular polygon that tessellates the plane or faces for a regular polyhedron that tessellates space, smallest order of a non-abelian group, triangular number, perfect number, etc.)

Edit:

Since you are presuming the numbers to be written in base 10 and looking for quick factor-checks, I’ll mention that you can test for divisibility by 11 much as you do for 9. Starting from the rightmost (ones) digit, alternately add and subtract each digit as you go left. The result is congruent to the input modulo 11. (This is made use of in the parlor trick for mentally taking cube roots of perfect cubes up to one billion in a few seconds.)

7 and 13 are a bit tougher, but you can do the same thing with alternating groups of three digits at a time to get below a thousand then brute-force it from there. The same idea but adding groups of three digits lets you reduce mod 37, and of course alternately adding and subtracting pairs of digits works for 101.
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Re: Interesting Numbers (Whole Numbers)

Postby Flumble » Tue Dec 30, 2014 9:11 pm UTC

I declare 3053 an interesting integer, because at the moment that you're reading this you don't know whether it is prime, nor do you know the chance that it is prime considering you now see that it's not divisible by three. Basically, you only know three (positive, odd and not a multiple of three) out of thousands of interesting properties of this number and that makes this number in particular very interesting.

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Re: Interesting Numbers (Whole Numbers)

Postby mike-l » Tue Dec 30, 2014 10:04 pm UTC

As a bit of a side note, you can check divisibility by any prime other than 2 or 5 using the following algorithm (which yields the normal results for 3 and 11).

Find a multiple of 10, 10m say that is one away from a multiple of your prime. The prime multiple can always be either the prime itself or 3 times it. By possibly taking m to be negative we may assume 10m is one more than a multiple of the prime. So eg for 7, m = -2, for 13, m = 4.

Starting from the right, multiply the final digit by m, and add this to the remainder of the number.

So eg, 2216214 is a multiple of both 7 and 13, which can be checked:

For 7

221621 -2*4 = 221613
22161 -2*3 = 22155
2215 -2*5 = 2205
220-2*5= 210
At this point it's evidently divisible by 7, but you can continue.

For 13
221621+4*4 = 221637
22163+4*7 = 22191
2219+4*1= 2223
222+4*3 = 234
23+4*4= 39
Which is divisible by 13

Of course m grows linearly with your prime, so this isn't much more useful than just straight up division.

For 3, m = 1 giving the usual 'add up the digits', for 11 m =-1 giving the usual alternating sum.


The downside of this is that it doesn't tell you the remainder if it isn't divisible, though it can be recovered with some extra work. Another method that does directly yield the remainder is to multiply the leftmost digit by the remainder when dividing 10 by your prime. This is only helpful for 7,13,17,19 as for larger primes the number gets bigger and the process doesn't terminate.

For example if we add 1 to the number above 2216215 we can find that remainder of 1 by repeatedly multiplying the leftmost digit by 3 and adding (for 7) or subtracting (for 13) it after moving it one place right:

216215+3*200000= 816215
16215+3*80000=256215
56215+3*20000=116215
16215+3*10000=46215
6215+3*4000=18215
8215+3*1000=11215
1215+3*1000=4215
215+3*400=1415
415+3*100=715
15+3*70=225
25+3*20=85
5+3*8=29
9+3*2=15
5+3*1=8
At any point you can stop when you can easily calculate the remainder. You can also simplify/speed up the process by removing 7s and multiples, so 8s become 1s, 43s become 01s, etc
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Re: Interesting Numbers (Whole Numbers)

Postby PM 2Ring » Wed Dec 31, 2014 2:00 am UTC

163 is rather interesting IMHO as it's the largest Heegner number; I guess I find it interesting because it's not immediately obvious that the set of Heegner numbers is finite, and also that it's such a small set.

All primes greater than 5 must be coprime to 2x3x5 = 30, IOW they must be of the form 30n + r where r is an element of {1, 7, 11, 13, 17, 19, 23, 29}; note that 1 + 29 = 7 + 23 = 11 + 19 = 13 + 17. The fact that that set has 8 members is handy for storing large tables of primes, since 1 byte can contain a bit pattern encoding all the primes in a block of 30 integers.

There are various tests that can be used to determine if a number is probably prime, eg Miller–Rabin. Composite numbers that "fool" such tests are kinda interesting, eg the Carmichael numbers; one rather famous Carmichael number is 1729, which is also the smallest natural number that can be expressed as the sum of two cubes in two different ways. This is sometimes known as the smallest non-trivial taxicab number.

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Re: Interesting Numbers (Whole Numbers)

Postby Forest Goose » Wed Dec 31, 2014 4:54 am UTC

196 is kind of neat, since it may be a Lychrel Number. It's not that I find the process itself particularly intriguing, but, for a lot of things of this type, you usually see "tested for all numbers up to x", where x is stupidly large. It's neat to see, "Well, it works up to 195".

Personally, I've always liked 7 - it is part of the only prime triplet (3, 5, 7), a Mersenne prime, every odd in it's Collatz trajectory is a twin prime, and some other stuff; but, really, it's just kind of cool on it's own, like a really awesome friend that you really aren't sure why they're awesome, but they just are.
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Re: Interesting Numbers (Whole Numbers)

Postby PM 2Ring » Wed Dec 31, 2014 5:17 am UTC

And 7 is the smallest cyclic prime in base 10. Generally, I'm not so interested in base-dependent properties of numbers - I consider them somewhat numerological, but I learned about the cyclic nature of 142857 when I was 9 or 10, so it has historical interest for me. :)


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