The City Registry of Boston records:
Benjamin Son of Josiah Frankling [sic] & Abiah his Wife born 6 January 1706
According to the Julian calendar then in use, Ben was born on January 6, 1706, but in 1752 Britain and its territories adopted the Gregorian calendar, ‘losing’ eleven days in the process, and moving New Year’s Day from March 25 to January 1, so we celebrate his birthday on January the 17th of 1706, although it is said that he always marked the 6th as his true birthdate. Incidentally, the Boston registry was later amended to reflect the new year designation, but not the new date.
Source: The Papers of Benjamin Franklin Yale University Press v.1 (1959) p.3