# Would the opposite of a perfect power be called a "root"?

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## Main Question or Discussion Point

I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their own, and they are the roots of their own integer powers.

I am actually exploring the Gaussian integers, and I assume these concepts apply equally to the Gaussian integers as they do to the rational integers.

Thanks!
-Jeffrey

Simon Bridge
Homework Helper
The solution to f(x)=0 is called a "root" of the equation, whether f(x) is a power or anything. It can also be called a "zero" of the equation.
If y = x^n, then we say that "x is the nth root of y" even if x is not an integer.

If x is an integer, then x is called a perfect nth root.

Thanks for the suggestion, Simon!

"Perfect nth root" makes sense. It's better than "root".

I would still like to find a way to express this class of numbers without having to reference an integer (i.e., "nth") of which it is a perfect root - since there are infinitely many n's. I think of these integers as analogous to the primes, except they are primitives of integer exponentiation instead of primitives of integer multiplication. Each of these integers is the first (x1) in an infinite series of perfect powers...and they are not members of any other set. If I'm not mistaken, the union of these sets is the set of all integers.

Simon Bridge
Homework Helper
There is a single-word name for a general perfect root ... it is called "integer".

Maybe I didn't explain myself. I'm talking about integers that do not have perfect roots.

mfb
Mentor
Maybe I didn't explain myself. I'm talking about integers that do not have perfect roots.
Every integer x has a perfect root $\sqrt{x}$, for example.

Hello mfb,

I believe you are incorrect.

You say that every integer has a perfect root. When you say "perfect" I assume that means integer. Of course every integer has roots, but not all of them are perfect (integers). The roots of integers are either integers or real (in fact, if they are not integers, they are actually irrational: https://proofwiki.org/wiki/Nth_Root_of_Integer_is_Integer_or_Irrational ).

"Is there a name for the class of integers that are not perfect powers?"

For instance:
x = 125 is a perfect power (it can be expressed as rn, where r = 5 and n = 3 (both integers > 1))
x = 25 is a perfect power (it can be expressed as rn, where r = 5 and n = 2 (both integers > 1))
But x = 5 is not a perfect power (it cannot be expressed as rn, where r and p are integers > 1.

The first entries of this unnamed set would be:
0, 1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28

The missing integers from this set are:
4, 8, 9, 16, 25, 27... (the perfect powers).

I am concluding that there is no official or agreed-upon name for this class of integers. Also, I cannot find that sequence in http://oeis.org. Maybe I'll add it.

Thanks,
-j

Mark44
Mentor
I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their own, and they are the roots of their own integer powers.
"Perfect powers" I've heard of, as 4 and 9 are perfect squares (of 2 and 3), and 64 and 125 are perfect cubes (of 4 and 5). I've not heard of the term "perfect root," or a distinction between a root of, say $x^2 = 16$ or $x^2 = 17$.
Ventrella said:
I am actually exploring the Gaussian integers, and I assume these concepts apply equally to the Gaussian integers as they do to the rational integers.
I suppose. For example, in the Gaussian integers, -1 = -1 + 0i is the square of 0 + i, and 1 = 1 + 0i is the fourth power or 0 + i. All of these are Gaussian integers.

mfb
Mentor
Ah, you require the root to be an integer.

So you are looking for the complement of oeis:A001597. "Not a perfect power" should do the job.

I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their own, and they are the roots of their own integer powers.
What you are looking for is: all the numbers for which the GCD (greater common divisor) of the exponents of all the powers of the primes in their prime decomposition is greater than 1. By definition the prime numbers (do not) belong in this list.

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What you are looking for is: all the numbers for which the GCD (greater common divisor) of the exponents of all the powers of the primes in their prime decomposition is greater than 1. By definition the prime numbers (do not) belong in this list.
Hi dagmar,

Could you elaborate? Perhaps you could decompose your definition so I can understand the component parts. Specifically, I don't understand: "...the exponents of all the powers of the primes in their prime decomposition..."

I don't see how prime numbers are involved here. But if you are correct that prime numbers are involved non-trivially, that would be interesting.

Thanks!
-j

Could you elaborate? Perhaps you could decompose your definition so I can understand the component parts. Specifically, I don't understand: "...the exponents of all the powers of the primes in their prime decomposition..."
Yup, I know my English is terrible,
$$n=p_1^{r_1} p_2^{r_2}....p_n^{r_n}$$ is prime decomposition of n. $$GCD(r_1,r_2,...,r_n)>1$$
GCD=2 => Number Perfect Square, GCD=3 => Perfect Cube, etc.

Sorry. Goodday.

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