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exponentiation - How to quickly identify perfect powers - Mathematics . . . Since you know that the first three can be written as an integer to an integer power you can eliminate those and focus on 4 and 5 Now, from here perhaps a factor tree can point you in the right direction Try quickly factoring 500 and if all factors are repeated more than once it is a perfect power If not, you found your answer
What better way to check if a number is a perfect power? The identity $(a^x)^y = a^{xy}$ tells us that it's sufficient to test only prime values of the exponent; if a number is a perfect power, it's also a perfect power with prime exponent (the base is different, of course) This lowers the number of tested exponents to eighteen
number theory - How many perfect powers are there amoung the first 1000 . . . If a number is a power then it is a power where the exponent is prime Furthermore there is no positive integer less than 1001 that is both (A) a square and a 5th or a 7th power B) a 3rd power and either a 5th or 7th power, (C) a 5th power and a 7th power (D) an 11th (or higher) power
terminology - Why are non-trivial powers called perfect powers . . . $\begingroup$ IMHO, perfect power or perfect square or the like is rather used as an emphasis: Surely $7$ is the quare of $\sqrt 7$, hence is "a square" - but it is not a perfect square Note that this is language-specific, e g in German perfect square is simply a "Quadratzahl" ("square number"), and the WP artcile "perfect power" links to a
Perfect numbers and powers of two - Mathematics Stack Exchange $\begingroup$ @user129901 A proof of Euler's result that all even perfect numbers are in fact of the form (the form that Euclid knew), can be found online several places, for example A proof that all even perfect numbers are a power of two times a Mersenne prime $\endgroup$ –
Proof for Perfect $n^{th}$ power. - Mathematics Stack Exchange Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
sum of reciprocal power-1 - Mathematics Stack Exchange Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
Rational perfect power tests - Mathematics Stack Exchange Hmmm, I think that your assertion that 4^0 5 is rational is arguable Bear in mind that in floating-point arithmetic 0 5 represents a range of real numbers, all of whose closest representation in the chosen version of f-p is closer to 0 5 than to either of its neighbouring f-p numbers
Newest perfect-powers Questions - Mathematics Stack Exchange A perfect power is a positive integer that can be expressed as an integer power of another positive integer This tag should only be used when having in mind an arbitrary perfect power (as opposed to a specific one, like a perfect square, a perfect cube, etc )