My Math Forum Monkey Root of $X^X$ numbers

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 March 7th, 2017, 02:35 AM #1 Senior Member   Joined: Dec 2012 Posts: 951 Thanks: 23 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.
March 7th, 2017, 03:02 AM   #2
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Quote:
 Originally Posted by complicatemodulus $\displaystyle -\sum_{X=1}^{X-1}X^X = X^X-\sum_{X=1}^{X}X^X$
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.

March 7th, 2017, 03:08 AM   #3
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Quote:
 Originally Posted by complicatemodulus 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.

March 7th, 2017, 05:19 AM   #4
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Quote:
 Originally Posted by v8archie 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....

March 7th, 2017, 06:54 AM   #5
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Quote:
 Originally Posted by complicatemodulus $\displaystyle R^-_n = -\sum_{X=1}^{X-1} X^X$
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

May 8th, 2017, 05:07 AM   #6
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Quote:
 Originally Posted by topsquark 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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