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Post a number with an interesting property

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Rudin
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08 Nov 2015, 3:52 pm

@SSV

Rudin wrote:
I said integers, so of course they can be negative (and 0). If they couldn't be negative I would specify.

1=(1+9n^3)^3+(9n^4)^3+(-9n^4-3n)^3

You can plug in any n you'd like. For instance,

1=1^3+0^3+0^3

And,

1=10^3+9^3-(12)^3

One is the only number that has been proven to have this property. You could expand out the parametric solution above and add it all up and you'd get one by the way.


Set n=1.


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SippingSpiderVenom
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08 Nov 2015, 5:15 pm

So n=1,
1000^3+729^3-1728^3 = 1

and
a = 1000^(1/3)=10
b = 729^(1/3)=9
c = 1728^(1/3)=12

I can also derive from n=2:
a = 144
b = 73
c = 150

n=3, becomes quite large


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09 Nov 2015, 10:41 pm

9.81 it is the metric rate of acceleration due to the gravitational pull of the earth. I think it's interesting because it is not 10 m/s^2 it's a glaring testament to the fact that we do not have a metric unit for time. So what do we do, throw out the metric system or the calendrical system? They are at first glance mutually exclusive.


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09 Nov 2015, 11:53 pm

22/7 is the fractional approximate for pi. I like it because I think it's easier to process in context - circumference over diameter, is contextually communicated in this form of pi.

Zero is interesting because it's a complete fabrication, there's no such thing as nothing, yet it's a remarkably useful tool. This actually makes it a complicated abstract construct which everyone understands quite easily. It all pans out mathematically as well when you try to divide by zero, nothing is composed of nothing and yet there it is, a fancy little round paradox sitting on your paper staring back at you.


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10 Nov 2015, 7:39 am

1=(5+((5+(5+(5+...)/6)/6)/6)/6

You can prove this because,

x=(5+x)/6

Therefore,

5x=5; x=1

Actually you can make a generalization, does anyone know what it is? If so, please share it.


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10 Nov 2015, 11:30 pm

Nope it's asymptotic though. Curiously if x=0 it equals 0, but if x = 1000 it sure wraps around one pretty quickly.


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Rudin
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11 Nov 2015, 7:16 am

Hint:

x=[(n-1)+x]/n


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11 Nov 2015, 11:32 pm

What is a generalization? Is this for proofs? I've only had algebra, enough to get a computer information systems degree. I've had some discrete maths where required and you need a certain amount of logic to learn how to program, but that's about it. I've read how to solve it, but I just set aside a formal logic class to concentrate on algebra. I'd like to learn higher math, but I'm stuck at sophomore level classes, so once I finish sophomore level maths I'll probably look at 200 level physics and chemistry, maybe computer science or engineering. At 3-4 classes a year it'll be about three years before I finish math anyway.

But if you'd like to teach me a thing here or there I'm willing to read and work a little bit. Do you program at all?


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Rudin
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12 Nov 2015, 7:55 am

SippingSpiderVenom wrote:
What is a generalization? Is this for proofs? I've only had algebra, enough to get a computer information systems degree. I've had some discrete maths where required and you need a certain amount of logic to learn how to program, but that's about it. I've read how to solve it, but I just set aside a formal logic class to concentrate on algebra. I'd like to learn higher math, but I'm stuck at sophomore level classes, so once I finish sophomore level maths I'll probably look at 200 level physics and chemistry, maybe computer science or engineering. At 3-4 classes a year it'll be about three years before I finish math anyway.

But if you'd like to teach me a thing here or there I'm willing to read and work a little bit. Do you program at all?


No, I don't program.

A generalization is what it sounds like.

So we have proved that 1 is a continued fraction involving 5 and 6. Since,

x=(5+x)/6

We could make a generalization :

x=(n-1+x)/n

nx=n-1+x

(n-1)x=(n-1)

x=1

Provided n is not 0, because you can't divide by zero. So the continued fraction works for any n not equal to zero, that's a generalization.

I should learn programming though, I tried C and made basic programs to calculate the length of the hypotenuse. This was around a year ago and I used books I signed out of the public library.

I would have continued, but I was too focused on maths.


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12 Nov 2015, 10:39 am

x=(n-1+x)/n

nx=n-1+x

(n-1)x=(n-1)

x=1


I could have done that. :oops:

I get,
nx-x=n-1

(nx-x)/x=(n-1)/x
n-1=(n-1)/x

and of course
(n-1)x=n-1
x=(n-1)/(n-1)
x=1

projecteuler.net is a cool site for programming/math problems. Maybe something to keep you busy over christmas break.


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Rudin
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13 Nov 2015, 7:31 am

SippingSpiderVenom wrote:
x=(n-1+x)/n

nx=n-1+x

(n-1)x=(n-1)

x=1


I could have done that. :oops:

I get,
nx-x=n-1

(nx-x)/x=(n-1)/x
n-1=(n-1)/x

and of course
(n-1)x=n-1
x=(n-1)/(n-1)
x=1

projecteuler.net is a cool site for programming/math problems. Maybe something to keep you busy over christmas break.


I'm sure you could. I was bored on my school bus, because it's 2 hours long, so I decided to do a little experimenting with continued fractions.

Thank you for the link. I will attempt some of these problems, to be honest I like the link you've sent me. I was actually looking for interesting problems like that.

Phi, which needs no introduction, is an awesome number.


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-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


Rudin
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13 Nov 2015, 7:33 am

1+1/x=x

x is phi so we could come up with some interesting continued fractions using that, I think there is already one.

It also shows up in almost all areas of mathematics. It also has an interesting relation with the Fibonacci sequence.


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"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

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13 Nov 2015, 7:35 am

Like this:

Image

We get x=sqrt(1+x) therefore,

x^2=1+x

x^2-x-1=0

Using the quadratic formula,

x=(1+sqrt(5))/2=phi


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13 Nov 2015, 8:05 am

145=1!+4!+5!


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"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider


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13 Nov 2015, 1:26 pm

5618

bonus points :nerdy: :nerdy: for the one who can tell me why.....WITHOUT using the internet :P :P :P :P



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13 Nov 2015, 9:52 pm

Gave up on it offline. And then I even went to the internet.

Believe it or not there is a whole Wiki type article on the net about the number 5618. Lists every characteristic about it. Its not a perfect number, its not a prime, its square root is nothing special. Nothin' special about it all.

I give up.