Question

A= 2014x (1+2+3+...+2013). Aratati ca A se poate scrie ca suma de 5 numere naturale , patrate perfecte, nenule si distincte

Answer 1

A= 2014 × (1+2+3+...+2013)=
=2014×2013×2014/2=
=1007×2×2013×1007=1007²×4026=
=60420² + 20140² + 4028² + 3021² + 1007²


4026 = 3600 + 400 +16+9 +1 ⇒ A = 1007² ·(60² +20² +4² +3²+1²) =
=(60×1007)² + (20×1007)² +(4×1007)² + (3×1007²) + 1007² =
= 60420² + 20140² + 4028² + 3021² + 1007²