Sum of powers
Contestcen
add up to the original number. 0 and 1 are trivial examples. The first
non-trivial example is 3435, since 3^3 + 4^4 + 3^3 + 5^5 = 27+64+27+3125 =
3435.
Can you find 2 more non-trivial examples?
Frank Rubin
fr...@contestcen.com
Aakash Mehendale
>
> A number is an auto-power number if each of its digits, taken to its own power,
> add up to the original number. 0 and 1 are trivial examples.
Well, at the risk of restarting old fires, 0^0 is generally taken to
be 1.
--
Aakash Mehendale email:aak...@popaccount.com
ICQ:26396017
Aleph-null Klein bottles, hanging on the wall...
Any views and opinions expressed herein are mine alone
Raycroft
(possibly) Starting a fire:
Take the function f(x,y)=x^y. We know from calculus that to be
continuous at A a function must have the same limit as x->A from both
directions (making them MUCH easier to disprove than prove). With more
than one variable, this extends to all paths, not just all directions.
Along the path x=0, we see f(x,y)=f(0,y)=0^y. Lim(y,0+)(0^y)=0. Along
the path y=0, we have f(x,y)=f(x,0)=x^0. Lim(x,0)(x^0)=1. therefore,
since the limits do not agree, the function is not continuous at the
origin, or 0^0. So while it may be convenient to use either 1 or 0 in
some cases, this is not a valid assumption. 0^0 is undefined.
Dave Seaman
Raycroft <k.ray...@worldnet.att.net> wrote:
>Aakash Mehendale wrote:
>> Well, at the risk of restarting old fires, 0^0 is generally taken to
>> be 1.
>(possibly) Starting a fire:
>Take the function f(x,y)=x^y. We know from calculus that to be
>continuous at A a function must have the same limit as x->A from both
>directions (making them MUCH easier to disprove than prove). With more
>than one variable, this extends to all paths, not just all directions.
>Along the path x=0, we see f(x,y)=f(0,y)=0^y. Lim(y,0+)(0^y)=0. Along
>the path y=0, we have f(x,y)=f(x,0)=x^0. Lim(x,0)(x^0)=1. therefore,
>since the limits do not agree, the function is not continuous at the
>origin, or 0^0. So while it may be convenient to use either 1 or 0 in
>some cases, this is not a valid assumption. 0^0 is undefined.
See the sci.math FAQ at
http://daisy.uwaterloo.ca/~alopez-o/math-faq/math-faq.html
The bottom line is that 0^0 is not a limit expression, and therefore
arguing that the limit is an indeterminate form has nothing to do with
whether the expression 0^0 has a defined value or not. Not every
function is continuous.
--
Dave Seaman dse...@purdue.edu
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal
<http://mojo.calyx.net/~refuse/altindex.html>
Mark Brader
>> A number is an auto-power number if each of its digits, taken to its
>> own power, add up to the original number. 0 and 1 are trivial
>> 2 more non-trivial examples?
Aakash Mehendale writes:
> Well, at the risk of restarting old fires, 0^0 is generally taken to
> be 1.
And in another branch of the thread, the sci.math FAQ list has been
cited for a good discussion. Rather than debating the point further
(I consider one of the three options to be obviously correct), I'd
merely like to point out that the choice might or might not affect
the puzzle.
That is, it's possible that the non-trivial examples that Frank intended
as answers assume 0^0 = 0; it's also possible that if you take 0^0 = 1,
you get solutions that Frank didn't have in mind. Or not; I haven't
checked. Of course, if you take 0^0 to be undefined, you will rule out
both types of solutions.
--
Mark Brader |"...one of the main causes of the fall of the Roman
Toronto | Empire was that, lacking zero, they had no way to
msbr...@interlog.com| indicate successful termination of their C programs."
| -- Robert Firth
My text in this article is in the public domain.
Jim Cook
Actually, 27+256+27+3125 = 3435
I couldn't find any four, five, or six digit numbers that fit the bill.
2792 is interesting in a similar way. 2^7+9^2=2792.
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Thomas
Mr. Klay I. Eno wrote in message <377dd380...@news.gate.net>...
>conte...@aol.com (Contestcen) wrote:
>whether you define 0^0 as being equal to zero, one, or undefined and
>therefore disallowed. I believe that 0^0 has no satisfactory definition.
>"Mr. Klay I. Eno" better known as 0627....@mail.serve.com.
> 01 2345 6 789 <- Use this key to decode my email address.
> Fun & Free - http://www.5X5poker.com/
I agree that there is no satisfactory definition for 0^0.
I was once almost autistically latched on f(x)=x^x, so interested was I in
the function. I asked my teachers and as far as I can remember I was told
that this at x=0 has no value, that is, it equals neither zero nor one.
Thomas
Mark Brader
> A number is an auto-power number if each of its digits, taken to its
> own power, add up to the original number. 0 and 1 are trivial
> examples. The first non-trivial example is 3435 ... Can you find
> 2 more non-trivial examples?
No, you can't. Unless I made a programming error#, there is only *one*
other example in base 10, and then only if you accept 0^0 = 0. This is
438,579,088 = 256+27+16777216+3125+823543+387420489+0+16777216+16777216.
# - I'll email the code on request; it's simple C, but ugly and repetitive.
Note that it is only necessary to search a finite number of integers, and
the limit conveniently fits in a 32-bit unsigned integer. n-digit positive
integers range from 10^n up to 10^(n+1)-1; the sum of n digits each raised to
its own power ranges from 0 or n (depending on how you interpret 0^0) up to
n*9^9.
Now 10^n defines a geometric progression while n*9^9 defines an arithmetic
one, so there exists N such that for all n >= N, 10^n > n*9^9. In partic-
ular, this is true for N = 11, so we only have to search numbers up to
10 digits. And then among 10-digit numbers we need go only up to 10*9^9,
which is 3,874,204,890, which is a 32-bit number in binary.
--
Mark Brader, Toronto "Remember that computers are very,
msbr...@interlog.com very fast..." -- Steve Summit
il...@isgtec.com
conte...@aol.com (Contestcen) wrote:
> A number is an auto-power number if each of its digits, taken to its
own power,
> add up to the original number. 0 and 1 are trivial examples. The
first
27+64+27+3125 =
>
> Can you find 2 more non-trivial examples?
>
> fr...@contestcen.com
>
SPOILER WARNING
If 0^0 is assumed to be zero, then 438,579,088 is such a number.
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John D. Goulden
power,
> add up to the original number. 0 and 1 are trivial examples. The first
> non-trivial example is 3435, since 3^3 + 4^4 + 3^3 + 5^5 = 27+64+27+3125 =
> 3435.
>
> Can you find 2 more non-trivial examples?
The only other one I can find is 438,579,088; that's assuming 0^0=0 for the
purposes of this problem. There are no solutions beyond 10*9^9.
--
Please reply by email as well as to the group.
John D. Goulden
jgou...@flash.net
Aakash Mehendale
>
[snip]
> See the sci.math FAQ at
>
> http://daisy.uwaterloo.ca/~alopez-o/math-faq/math-faq.html
>
> The bottom line is that 0^0 is not a limit expression, and therefore
> arguing that the limit is an indeterminate form has nothing to do with
> whether the expression 0^0 has a defined value or not. Not every
> function is continuous.
>
Yes. It was that particular paragraph from Concrete Maths that had
stuck in my mind.