He couldn't check each possible method and maybe there were very involved methods that didn't seem to have anything to do with Diophantine equations but still worked.
Siya ay hindi suriin ang bawat posibleng paraan at marahil ay may tunay na kasangkot pamamaraan na hindi mukhang na magkaroon ng kahit na ano ang gagawin sa Diophantine equation pero nagtrabaho pa rin.