Inmathematics, aperfect power is anatural number that is a product of equal natural factors, or, in other words, aninteger that can be expressed as a square or a higher integerpower of another integer greater than one. More formally,n is a perfect power if there existnatural numbersm > 1, andk > 1 such thatm^k =n. In this case,n may be called aperfectkth power. Ifk = 2 ork = 3, thenn is called aperfect square orperfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers (0^k = 0 for anyk > 0, 1^k = 1 for anyk).

Asequence of perfect powers can be generated by iterating through the possible values form andk. The first few ascending perfect powers in numerical order (showing duplicate powers) are (sequenceA072103 in theOEIS):

${\displaystyle 2^2=4,2^3=8,3^2=9,2^4=16,4^2=16,5^2=25,3^3=27,}$${\displaystyle 2^5=32,6^2=36,7^2=49,2^6=64,4^3=64,8^2=64,\dots }$

Thesum of thereciprocals of the perfect powers (including duplicates such as 3⁴ and 9², both of which equal 81) is 1:

${\displaystyle \sum _{m=2}^{\mathrm{\infty }}\sum _{k=2}^{\mathrm{\infty }}\frac{1}{m^k}=1.}$

which can be proved as follows:

${\displaystyle \sum _{m=2}^{\mathrm{\infty }}\sum _{k=2}^{\mathrm{\infty }}\frac{1}{m^k}=\sum _{m=2}^{\mathrm{\infty }}\frac{1}{m^2}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{m^k}=\sum _{m=2}^{\mathrm{\infty }}\frac{1}{m^2}\left(\frac{m}{m-1}\right)=\sum _{m=2}^{\mathrm{\infty }}\frac{1}{m(m-1)}=\sum _{m=2}^{\mathrm{\infty }}\left(\frac{1}{m-1}-\frac{1}{m}\right)=1.}$

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, ... (sequenceA001597 in theOEIS)

The sum of the reciprocals of the perfect powersp without duplicates is:^[1]

${\displaystyle \sum _p\frac{1}{p}=\sum _{k=2}^{\mathrm{\infty }}\mu (k)(1-\zeta (k))\approx 0.874464368\dots }$

where μ(k) is theMöbius function and ζ(k) is theRiemann zeta function.

According toEuler,Goldbach showed (in a now-lost letter) that the sum of⁠1/p − 1⁠ over the set of perfect powersp, excluding 1 and excluding duplicates, is 1:

${\displaystyle \sum _p\frac{1}{p-1}=\frac{1}{3}+\frac{1}{7}+\frac{1}{8}+\frac{1}{15}+\frac{1}{24}+\frac{1}{26}+\frac{1}{31}+\cdots =1.}$

This is sometimes known as theGoldbach–Euler theorem.

Detecting whether or not a given natural numbern is a perfect power may be accomplished in many different ways, with varying levels ofcomplexity. One of the simplest such methods is to consider all possible values fork across each of thedivisors ofn, up to${\displaystyle k\le {\log }_{2}n}$. So if the divisors of${\displaystyle n}$ are${\displaystyle n_1,n_2,\dots ,n_j}$ then one of the values${\displaystyle n_1^2,n_2^2,\dots ,n_j^2,n_1^3,n_2^3,\dots }$ must be equal ton ifn is indeed a perfect power.

This method can immediately be simplified by instead considering onlyprime values ofk. This is because if${\displaystyle n=m^k}$ for acomposite${\displaystyle k=ap}$ wherep is prime, then this can simply be rewritten as${\displaystyle n=m^k=m^{ap}=(m^a{)}^p}$. Because of this result, theminimal value ofk must necessarily be prime.

If the full factorization ofn is known, say${\displaystyle n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r}}$ where the${\displaystyle p_i}$ are distinct primes, thenn is a perfect powerif and only if${\displaystyle gcd({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_r)>1}$ where gcd denotes thegreatest common divisor. As an example, considern = 2⁹⁶·3⁶⁰·7²⁴. Since gcd(96, 60, 24) = 12,n is a perfect 12th power (and a perfect 6th power, 4th power, cube and square, since 6, 4, 3 and 2 divide 12).

In 2002 Romanian mathematicianPreda Mihăilescu proved that the only pair of consecutive perfect powers is 2³ = 8 and 3² = 9, thus provingCatalan's conjecture.

Pillai's conjecture states that for any given positive integerk there are only a finite number of pairs of perfect powers whose difference isk. This is an unsolved problem.^[2]

• Prime power

• Daniel J. Bernstein (1998)."Detecting perfect powers in essentially linear time"(PDF).Mathematics of Computation.67 (223):1253–1283.doi:10.1090/S0025-5718-98-00952-1.

• Lluís Bibiloni, Pelegrí Viader, and Jaume Paradís, On a Series of Goldbach and Euler, 2004 (Pdf)

• Number theory
• Integer sequences

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