Section 3. For all elections and primaries held prior to June first in any year, a person shall be registered and may vote in the ward or voting precinct where he resided on January first of the previous year; provided, however, that any registered voter of a city or town who moves to any other precinct in said city or town may register to vote at his new address by making written application to the city or town clerk no later than the close of registration. A new resident of the city or town who changes his residence after having first registered in the city or town may also, upon like application, be registered at the new address. For all elections and primaries held on or after June first in any year, a person shall be registered and may vote in the ward or voting precinct where he resided on January first of that year; provided, however, that any registered voter of a city or town who moves to any other precinct in said city or town may register to vote at his new address by making written application to the city or town clerk no later than the close of registration. A new resident of the city or town who changes his residence after having first registered in the city or town may also, upon application, be registered at the new address. The city or town clerk shall forthwith notify each voter making any such written application that the same has been received and that he may vote, subject to the provision of this section regarding the close of registration, in the ward or precinct into which he has moved. The registrars may require that any such application be signed under the penalties of perjury.