Representation of positive integers by the form x³+y³+z³-3xyz

الأصل · EN

We study the number ν(n) of representations of a positive integer n by the form x³+y³+z³-3xyz in the conditions 0≤ x≤ y≤ z; z≥ x+1. We proved the following results: (i) for every positive n, except for n≡±3 9, ν(n)>=1; (ii) for the exceptional n, ν(n)=0; (iii) for every prime p≠3, ν(p)=ν(2p)=1; (iv) (ν(n))=∞; (v) for every positive n, there exists k such that ν(k)=n.