Talk:Perfect power

Disambiguation

This entry needs disambiguation from the Galvin Institute's Perfect Power, which is all about better electrical power. —Preceding unsigned comment added by 134.134.139.70 (talk)

Wikipedia is not a dictionary attempting to list all possible meanings of a term. Wikipedia:Disambiguation#Deciding to disambiguate says: Disambiguation is required whenever, for a given word or phrase on which a reader might use the "Go button", there is more than one Wikipedia article to which that word or phrase might be expected to lead. There doesn't appear to be information about Galvin Institute's Perfect Power in Wikipedia, so no disambiguation is needed. PrimeHunter (talk) 21:52, 20 November 2009 (UTC)

Computer Program for Perfect Powers

Has anyone figured out a computer program for determining perfect powers in chronological order (A001597), excluding 1? Where n is the instance in the set and y is the value of that instance:

n=1, y=4
n=2, y=8
n=3, y=9
n=4, y=16
n=5, y=25
n=6, y=27
n=7, y=32
n=8, y=36

What about a math equation? Jeanlovecomputers (talk) 01:19, 14 February 2010 (UTC)

It's not too hard to come up with an algorithm for producing an ordered list of perfect powers:

1. Start with a set S containing 2² = 4, and an empty list L.
2. Find the smallest member of S - call it x. Then x is n^m for some n and m.
3. Remove x from S and append x to L.
4. Add n^m+1 to the set S.
5. If m is even then x is a square - suppose x = k². Add (k+1)² to the set S also.
6. Repeat from 2.

So the first few iterations are:

L = {}, S = {4}. Remove 4 from S and append it to L. Add 2³ = 8 and 3² = 9 to S.

L = {4}, S = {8,9}. Remove 8 from S and append it to L. Add 2⁴ = 16 to S.

L = {4,8}, S = {9,16}. Remove 9 from S and append it to L. Add 3³ = 27 and 4² = 16 to S.

L = {4,8,9}, S = {16,27}. Remove 16 from S and append it to L. Add 2⁵ = 32 and 5² = 25 to S.

L = {4,8,9,16}, S = {27,32,25}. Remove 25 from S and append it to L. Add 5³ = 125 and 6² = 36 to S.

L = {4,8,9,16,25}, S = {27,32,125,36}. Remove 27 from S and append it to L. Add 3⁴ = 81 to S.

L = {4,8,9,16,25,27}, S = {32,125,36,81}. Remove 32 from S and append it to L. Add 2⁶ = 64 to S.

etc. Gandalf61 (talk) 10:10, 14 February 2010 (UTC)

I did something of the like to create a Qbasic program. All my program asks for is a maximum y. It scans each integer base for each integer $\geq 4$ to y. The maximum m, which is the base, is:

$\lfloor {\sqrt {y}}\rfloor$

I wish I could put up a screenshot of my program. But again, is there an equation for this? Because equations can't have booleans, can they? Jeanlovecomputers (talk) 15:46, 14 February 2010 (UTC)

New information: The set A001597 without the 1 is the set A072777. Jeanlovecomputers (talk) 20:41, 16 February 2010 (UTC)

Set A072777

Here is a mathematical representation of the set:

$P=\lbrace m^{k}:m,k\in \mathbb {N} \,,m,k>1\rbrace$

Jeanlovecomputers (talk) 20:39, 30 April 2010 (UTC)

Duplicated summation symbols?

In "Examples and sums", I think some of the summation symbols on the left are mistakenly duplicated. — Preceding unsigned comment added by Joseki546 (talk • contribs) 02:29, 8 May 2020 (UTC)

No, they're correct. For example, $\sum _{m=2}^{\infty }\sum _{k=2}^{\infty }{\frac {1}{m^{k}}}$ can be read like this: For all numbers $m=2..{\infty }$ (that's the first summation symbol), calculate all their powers for exponents $k=2..{\infty }$ (that's the second summation symbol) and sum the reciprocals of the results.

Maybe it becomes clearer when we write it without summation symbols: ${\frac {1}{2^{2}}}+{\frac {1}{2^{3}}}+{\frac {1}{2^{4}}}+...+{\frac {1}{3^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{3^{4}}}+...+{\frac {1}{4^{2}}}+{\frac {1}{4^{3}}}+{\frac {1}{4^{4}}}+...+...$ Chrisahn (talk) 09:24, 8 May 2020 (UTC)