In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k).
Examples and sums A sequence of perfect powers can be generated by iterating through the possible values for m and k. The first few ascending perfect powers in numerical order (showing duplicate powers) are (sequence A072103 in the OEIS):
2 2 = 4 , 2 3 = 8 , 3 2 = 9 , 2 4 = 16 , 4 2 = 16 , 5 2 = 25 , 3 3 = 27 , {\displaystyle 2^{2}=4,\ 2^{3}=8,\ 3^{2}=9,\ 2^{4}=16,\ 4^{2}=16,\ 5^{2}=25,\ 3^{3}=27,} 2 5 = 32 , 6 2 = 36 , 7 2 = 49 , 2 6 = 64 , 4 3 = 64 , 8 2 = 64 , … {\displaystyle 2^{5}=32,\ 6^{2}=36,\ 7^{2}=49,\ 2^{6}=64,\ 4^{3}=64,\ 8^{2}=64,\dots }
The sum of the reciprocals of the perfect powers (including duplicates such as 34 and 92, both of which equal 81) is 1:
∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = 1. {\displaystyle \sum _{m=2}^{\infty }\sum _{k=2}^{\infty }{\frac {1}{m^{k}}}=1.}
which can be proved as follows:
∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 ∑ k = 0 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 ( m m − 1 ) = ∑ m = 2 ∞ 1 m ( m − 1 ) = ∑ m = 2 ∞ ( 1 m − 1 − 1 m ) = 1 . {\displaystyle \sum _{m=2}^{\infty }\sum _{k=2}^{\infty }{\frac {1}{m^{k}}}=\sum _{m=2}^{\infty }{\frac {1}{m^{2}}}\sum _{k=0}^{\infty }{\frac {1}{m^{k}}}=\sum _{m=2}^{\infty }{\frac {1}{m^{2}}}\left({\frac {m}{m-1}}\right)=\sum _{m=2}^{\infty }{\frac {1}{m(m-1)}}=\sum _{m=2}^{\infty }\left({\frac {1}{m-1}}-{\frac {1}{m}}\right)=1\,.}
The first perfect powers without duplicates are:
(sometimes 0 and 1), 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, ... (sequence A001597 in the OEIS) The sum of the reciprocals of the perfect powers p without duplicates is:
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