Doua siruri care nu au nici un termen divizibil cu 2003

[Vlad Matei]

Fie \( \displaystyle (a_{n})_{n\geq 0},(b_{n}) _{n\geq 0} \) doua siruri de numere naturale astfel incat \( a_{0}=1,b_{0}=4 \) si \( a_{n+1}=a_{n} ^{2001}+b_{n} \), \( b_{n+1}=b_{n} ^{2001}+a_{n} \). Demonstrati ca \( (2003, a_{n})=1 \) si \( (2003, b_{n})=1 \).
(IBMO 2003)