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Cook made 136 not out as England reached 216-1 in reply to India's 316 all out at the close of day two in Kolkata, the BBC reports.
In doing so, the opening batsman went past the mark of 22 shared by Wally Hammond, Colin Cowdrey, Geoffrey Boycott and Kevin Pietersen.
His opening partner Nick Compton was out for 57, and with him at the wicket is Jonathan Trott on 21.
Cook also became the youngest batsman to reach 7000 test runs, reaching the milestone at the age of 27 years and 347 days, to knock Sachin Tendulkar off top spot.
Tendulkar, who has compiled a record 15,638 test runs, was 28 years and 193 days old when he passed the 7000 mark.
Cook is now joint-20th on the list of test centurions headed by Tendulkar with 51.