Refined prime power and perfect power checkers

Author: jakewilliami

.gitignore

@@ -28,3 +28,4 @@ Manifest.toml
 
 # miscellaneous stuff
 other/
+PRIME_README/

Project.toml

@@ -5,9 +5,9 @@ version = "0.2.3"
 
 [deps]
 FLoops = "cc61a311-1640-44b5-9fba-1b764f453329"
+Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Mmap = "a63ad114-7e13-5084-954f-fe012c677804"
-Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Polynomials = "f27b6e38-b328-58d1-80ce-0feddd5e7a45"
 Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"

src/CodingTheory.jl

@@ -6,7 +6,8 @@
 
 module CodingTheory
 
-using Nemo: isprime, factor, fmpz
+# using Nemo: isprime, factor, fmpz # for prime power function
+using Hecke: ispower, isprime_power # for perfect power function
 # using Primes: isprime, primes
 using LinearAlgebra: I
 using FLoops: @floop, ThreadedEx

src/powers.jl

@@ -1,3 +1,43 @@
+# Perfect power function
+
+@doc raw"""
+```julia
+isperfectpower(n::Integer) -> Bool
+```
+
+Given an integer `n`, returns `true` or `false` depending on whether or not it is a perfect power.
+
+Here, a perfect power is some number of the form ``a^b``, where ``a, b \in \mathbb{N}``, and ``b > 1``.
+
+This function is a wrapper around [Hecke.jl's really excellent and efficient `ispower` function](https://github.com/thofma/Hecke.jl/blob/master/src/Misc/Integer.jl#L413-L443).
+
+!!! note
+	
+	It should be noted that there is another defintion for "perfect powers":
+		> some number of the form ``p^b``, where ``p, b \in \mathbb{N}``, ``b > 1``, *and ``p`` is a prime number``.
+	Here, we call numbers of this form *prime powers*, or *proper perfect powers*.  By this definition, 36 is *not* a perfect power, because 6 is not prime.
+"""
+function isperfectpower(n::Integer)
+    (n == 0) && return false
+    e = first(Hecke.ispower(n))
+    return (e != 1 && e != 0)
+end
+
+"""
+```julia
+isprimepower(n::Integer) -> Bool
+```
+
+This function is a wrapper around [Hecke.jl's really excellent and effcient `ispower` function](https://github.com/thofma/Hecke.jl/blob/master/src/Misc/Integer.jl#L756-L769).
+
+"""
+function isprimepower(n::Integer)
+    return isprime_power(n)
+end
+
+## Prime power function
+
+#=
 function __isprime(n)
 	return isprime(fmpz(n))
 end
@@ -39,10 +79,14 @@ function Ω(n)
     return sum(e for (__, e) ∈ __factors(n))
 end
 
+# this is a slow version compared to the Hecke one
 function isperfectpower(n)
 	return ω(n) == 1 && Ω(n) != 1
 end
 
+# this is slower than the Hecke verson
 function isprimepower(n)
     return ω(n) == 1
 end
+
+=#

src/utils.jl

@@ -55,6 +55,10 @@ Check that all elements in a list are equal to each other.
 end
 @inline allequal(a::T...) where {T} = allequal(a)
 
+@inline function allequal2(A)
+	return !A=A[1:1]==A∪A
+end
+
 """
 ```julia
 aredistinct(A) -> Bool

src/wip/primes.jl

@@ -92,8 +92,8 @@ function perfectpower_brute_force(n::Integer)
 	n ≤ 1 && return nothing
 
 	for a in 1:n
-		for m in 1:n
-			isequal(a^m, n) && return a, m
+		for m in 2:n
+			isequal(a^m, n) && return m, a
 		end
 	end