Uniqueness of pair $\left(a,b\right)$ in writing positive integer $V$ as $V=a^2+ab+b^2$ with $a, b \in \mathbb{N}$

It's not unique. $$(a,b)=(6,5)\implies a^2+ab+b^2=36+30+25=91$$

$$(a,b)=(9,1)\implies a^2+ab+b^2=81+9+1=91$$

Tags:

Number Theory

Abstract Algebra

Diophantine Equations

Related

Proving the identity $\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$ by considering a more general form Why do we divide Permutations to get to Combinations? Smallest $1\ 000$ digit number $x$ , such that $x$ , $x+2$ and $x+6$ are all prime? Sum to infinity of a Series Question related to Pascal Triangles and Counting Patterns Linear algebra: cross product property Finding a basis of an infinite dimensional vector space with a given vector Why am I running into problems when I don't fully reduce this integral problem? Ramanujan's radical and how we define an infinite nested radical Are there non-trivial examples of an 'infinitieth' derivative? What is the criteria for 'convergence'? Prove that a cyclic group with only one generator can have at most 2 elements In an isosceles triangle $ABC$ show that $PM+PN$ does not depend on the position of the chosen point P.

Recent Posts