It is highly recommended to make a more efficient search by ticket number (if its a ticket what you are looking for of course

to change the default search parameters go to:
SysConfig -> Ticket -> Frontend::Agent::Ticket::ViewSearch
Look for: Ticket::Frontend::AgentTicketSearch###Defaults###Fulltext
deactivate the checkbox and activate :
Ticket::Frontend::AgentTicketSearch###Default###TicketNumber
(you can choose any other default option this way)
Update.
This will leave the Search By Ticket Number as default, leaving the option to search in the whole DB IF needed.
This will prevent your agents for making ticket searches in the WHOLE database, thus incrementing efficiency and resources.