Question

$\sqrt{2007×2009×2015×2017+64}$ ∈ N

Answer 1

2007×2009×2015×2017+64 =
= (2012-5)(2012-3)(2012+3)(2012+5)+64 =
= (2012-5)(2012+5)(2012-3)(2012+3)+64 =

[Notăm 2012 = x]

= (x-5)(x+5)(x-3)(x+3)+64 =
= (x²-5²)(x²-3²)+64 =
= (x²-25)(x²-9)+64 =
= x⁴-9x²-25x²+9×25+64 =
= x⁴-34x²+289 =
= x⁴-34x²+17²=
= (x²-17)² =

[revenim la notație]

= (2012²-17)²

=> √( (2012²-17)² ) = 2012²-17 număr natural.