Hensel's lemma the value of "c" mod 27: if "c" ≡ 1 mod 27 then use "a" = 1, if "c" ≡ 10 mod 27 then use "a" = 4 (since 4 is a root of "f"("x") mod 27), and if "c" ≡ 19 mod 27 then use "a" = 7. (It is not true that every "c" ≡ 1 mod 3 is a 3-adic cube, e.g., 4 is not a 3-adic cube since it is not a cube mod 9.) In a similar way, after some preliminary work Hensel's lemma can be used to show that for any "odd" prime number
Doubling the cube was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths and , the duplication of the cube is equivalent to finding segments of lengths and so that In turn, this means that But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass. Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the