No. The Constitution makes no mention of Presidents having to live in the White House. In fact the White House and Washington D.D. for that matter did not even exist when the Constitution was adopted.
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All have, except George Washington, though for most of his second term Harry Truman lived across the street in Blair House while the White House was being rebuilt.
The White House Office
President's residence
President's residence
The White House is known as the nerve center of the executive office of the president. James Hoban was the architect of the White House.
The park that is located to the north of the White House is Lafayette Square.