# [seqfan] I searched for: n^(n-1) = (n-1)^n and n^(n+1) = (n+1)^n

Daniel Forgues squid at zensearch.com
Thu Jun 17 18:28:24 CEST 2010

```I searched for:

n^(n-1) = (n-1)^n
n^(n+1) = (n+1)^n

with Wolfram Alpha:

[http://www.wolframalpha.com/input/?i=n^%28n-1%29+%3D+%28n-1%29^n n^(n-1) = (n-1)^n]
[http://www.wolframalpha.com/input/?i=n^%28n%2B1%29+%3D+%28n%2B1%29^n n^(n+1) = (n+1)^n]

and got:

n ~~ 3.293166287411861031508028291250805864372257290327121...
n ~~ 2.293166287411861031508028291250805864372257290327121...

which have the same decimal expansions (only offset of 1, obviously expected.)

I checked:

[http://oldweb.cecm.sfu.ca/cgi-bin/isc/lookup?number=3.293166287411861031508028291250&lookup_type=simple ISC]

[http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=2.293166287411861031508028291250&lookup_type=simple Plouffe's Inverter]

[http://www.research.att.com/~njas/sequences/?q=293166287411861031508028291250805864372&sort=0&fmt=0&language=english&go=Search OEIS]

And

n ~~ 3.293166287411861031508028291250805864372257290327121...
n ~~ 2.293166287411861031508028291250805864372257290327121...

are not in ISC, Plouffe's Inverter, OEIS...

Are these 2 related equations with their related (probably transcendental...) constants mathematically insignificant/uninteresting? Anybody ever looked for this?  I didn't find anything online about it, is it worth investigating?

I just happened to like the symmetrical form of those equations:

n^(n-1) = (n-1)^n
n^(n+1) = (n+1)^n

Thanks

Daniel Forgues

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