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I found some numbers that are really interesting!
n:number of digits
n=3,
1^3+5^3+3^3=153
3^3+7^3+0^3=370
3^3+7^3+1^3=371
4^3+0^3+7^3=407
n=4,
8^4+2^4+0^4+8^4=8208
9^4+4^4+7^4+4^4=9474
1^4+6^4+3^4+4^4=1634
n=5,
92727
54748
93084
n=6,
548834
n=7,
9926315
Last edited by wcy (20120113 18:28:33)
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Last edited by wcy (20120113 18:29:06)
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I don't know if these two count because the power isn't the number of digits, but 4150= 4^5+1^5+5^5+0^5 and 4151= 4^5+1^5+5^5+1^5
There are a few more where n=7: 1,741,725; 4,210,818; 9,800,817
There's an n=8: 24,678,050
n=9: 146,511,208; 472,335,975; 534,494,836; 912,985,153
n=10: 4,679,307,774
A bit different: 3435= 3^3+4^4+3^3+5^5 and 40585= 4!+0!+5!+8!+5!
And for the grand finale, wait, this is too amazing to just put here. I'll put it in this
Last edited by mathsyperson (20050804 02:46:17)
Why did the vector cross the road?
It wanted to be normal.
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mathsy! How did you do it!
I was thinking along the lines:
Start with: 1^3+5^3+3^3=153
Expand decimal: 1^3+5^3+3^3=1×10^2 + 5×10^1 + 3×10^0
Algebraically: a^3+b^3+b^3=a×10^2 + b×10^1 + c×10^0
But that was as far as I got ...
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
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Superlative work, Mathsy
How did you do it?
Character is who you are when no one is looking.
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Sorry to disappoint everyone, but I kind of cheated...
Why did the vector cross the road?
It wanted to be normal.
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Where did you cheat? I can see the website, but not your answers...
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CHEATING!!!!
How dare you!!!
Even though we all do it.
Friends are angels who lift our feet when our own wings have trouble remembering how to fly
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Scroll down to just below halfway and they're in a big black box.
Why did the vector cross the road?
It wanted to be normal.
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Oh right. I see. Tut tut, Mathsy. You shouldn't have told us...
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Bad boy!!
Friends are angels who lift our feet when our own wings have trouble remembering how to fly
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Ummm ... mathsy was not cheating when he tells us, right?
It comes under the heading "havin' a little fun with us" I think.
Good find, by the way. Interesting website, too.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
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wow this is cool!!
i was stuck at n=8, where there were 100 million possibilities.. i was using Javascript, and my computer sort of died..
now for the ultimate question:
Prove, using induction or otherwise, that there are solutions for all n element of Z+, n>2.
(i don't know the answer myself )
Last edited by wcy (20050804 23:26:14)
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For a computerbased solution you may be better off using a faster language like C, Java or .Net. But before that, there may be some tricks to speed it up  for example, there may be whole ranges that can be excluded from the search. (But I didn't look at your code, so maybe you are already doing that.)
So, are we back to a^3 + b^3 + c^3 = a×10^2 + b×10^1 + c×10^0 ?
This can be rearranged to: a^3 + b^3 + c^3  a×10^2  b×10^1  c×10^0 = 0
Then: a^3  a×10^2 + b^3  b×10^1 + c^3  c×10^0 = 0
Then: a(a^2  10^2) + b(b^2  10^1) + c(c^2  10^0) = 0
Is this helping or making it worse I wonder.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
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That is, it is to be proved that
(abcd..........)^n = a^n+b^n+c^n+d^n..................
Here, both a,b,c,d...etc and n go on and on.
It is easy to say there would always exist a n digit number whose sum of digits raised to the power n is equal to the number. To prove that may be quite difficult. As n becomes higher, the combinations available increase, thereby increasing the possibility of such a number existing.
Character is who you are when no one is looking.
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I think that as there is no answer for n=2, there are probably other values of n for which there are no answers. For this reason, I think it would be better to prove by counterexample. Of course, we'll need 10^n counterexamples, but it's still easier than proving something that's not true.
If you still want to try, then don't be discouraged by my scepticism. After all, Fermat's Last Theorem doesn't work with n=2 either!
Why did the vector cross the road?
It wanted to be normal.
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