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March 7th, 2017, 01:35 AM  #1 
Senior Member Joined: Dec 2012 Posts: 1,022 Thanks: 24  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: $P1^1 = R_1$ $R_12^2=R_2$ ... till you've the first negative value $R^_n$. If the negative value is equal to: $\displaystyle R^_n = \sum_{X=1}^{X1} X^X$ than $P=X^X$ It require only n step example: $P= 27$ $R_1=271^1=26$ $R_2=262^2=22$ $R_3=223^3=5$ $1^1+2^2 = 5 = R_3 $ than $P=27=3^3$ From the trivial identity: $\displaystyle \sum_{X=1}^{X1}X^X = X^X\sum_{X=1}^{X}X^X $ Last edited by complicatemodulus; March 7th, 2017 at 01:37 AM. 
March 7th, 2017, 02:02 AM  #2  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,878 Thanks: 2240 Math Focus: Mainly analysis and algebra  Quote:
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 02:59 AM.  
March 7th, 2017, 02:08 AM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,290 Thanks: 441 Math Focus: Yet to find out.  How to test this if I don't know what X is?
Last edited by skipjack; March 7th, 2017 at 09:09 AM. 
March 7th, 2017, 04:19 AM  #4  
Senior Member Joined: Dec 2012 Posts: 1,022 Thanks: 24  Quote:
 
March 7th, 2017, 05:54 AM  #5 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,570 Thanks: 613 Math Focus: Wibbly wobbly timeywimey stuff.  
May 8th, 2017, 05:07 AM  #6 
Senior Member Joined: Dec 2012 Posts: 1,022 Thanks: 24  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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$xx$, monkey, numbers, root 
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