Question

Determinati numerele naturale r cu proprietatea ca exista numere naturale prime p si q, astfel incat p²+pq+q²=r²

Answer 1

p²+pq+q²=r²⇔p²+2pq+q²-pq=r²⇔(p+q)²-r²=pq⇔(p+q-r)(p+q+r)=pq
Cum p si q sunt prime si p+q+r> max ( p,q ), obtinem ca p+q-r=1 si p+q+r=pq
Adunand relatiile, obtinem pq+1=2(p+q)⇔pq-2p-2q+4=3⇔(p-2)(q-2)=3
Obtinem (p,q)∈{(3,5),(5,3)}⇒r=7