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March 7th, 2017, 02:35 AM   #1
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Monkey Root of $X^X$ numbers

To know if a number $P\in\mathbb{N^+}$ is of the form $X^X$

you can make the recoursive difference from P of:

$P-1^1 = R_1$

$R_1-2^2=R_2$
...

till you've the first negative value $R^-_n$.

If the negative value is equal to:

$\displaystyle R^-_n = -\sum_{X=1}^{X-1} X^X$

than $P=X^X$

It require only n step

example:

$P= 27$

$R_1=27-1^1=26$

$R_2=26-2^2=22$

$R_3=22-3^3=-5$

$1^1+2^2 = 5 = -R_3 $

than $P=27=3^3$

From the trivial identity:

$\displaystyle -\sum_{X=1}^{X-1}X^X = X^X-\sum_{X=1}^{X}X^X $


Last edited by complicatemodulus; March 7th, 2017 at 02:37 AM.
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March 7th, 2017, 03:02 AM   #2
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Originally Posted by complicatemodulus View Post
$\displaystyle -\sum_{X=1}^{X-1}X^X = X^X-\sum_{X=1}^{X}X^X $
I don't know what you think that this represents.

Your example shows you subtracting $1^1$, $2^2$ and $3^3$ from 27. If I already know $3^3$, why do I need your process? It is quicker to calculate $k^k$ for $k=1,2,\ldots$ until $k^k \ge p$ with equality if $k^k = p$.

Also, calculating $k^k$ is order $k$, so the algorithm is order $k^k$.

Last edited by v8archie; March 7th, 2017 at 03:59 AM.
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March 7th, 2017, 03:08 AM   #3
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Originally Posted by complicatemodulus View Post
T

$\displaystyle R^-_n = -\sum_{X=1}^{X-1} X^X$
How to test this if I don't know what X is?

Last edited by skipjack; March 7th, 2017 at 10:09 AM.
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March 7th, 2017, 05:19 AM   #4
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Quote:
Originally Posted by v8archie View Post
I don't know what you think that this represents.

Your example shows you subtracting $1^1$, $2^2$ and $3^3$ from 27. If I already know $3^3$, why do I need your process? It is quicker to calculate $k^k$ for $k=1,2,\ldots$ until $k^k \ge p$ with equality if $k^k = p$.

Also, calculating $k^k$ is order $k$, so the algorithm is order $k^k$.
I suppose "Monkey" in the title was enough....
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March 7th, 2017, 06:54 AM   #5
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Originally Posted by complicatemodulus View Post
$\displaystyle R^-_n = -\sum_{X=1}^{X-1} X^X$
This expression makes no sense! Is X a summation variable or a number?

What does $\displaystyle \sum _{5 = 1}^{5-1} 5^5$ mean?

-Dan
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May 8th, 2017, 05:07 AM   #6
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Quote:
Originally Posted by topsquark View Post
This expression makes no sense! Is X a summation variable or a number?

What does $\displaystyle \sum _{5 = 1}^{5-1} 5^5$ mean?

-Dan
Sorry, I forgot to reply... "x" is the variable in the Sum, "X" the value / uppervalue... I hope was clear and "monkey" was enough to say that is better to write $X^X$ as $A^A$ or any other character to avoid missunderstanding like to one here...
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