No single agreed answer; Egypt, Iran, China, Japan contend
There is no single agreed answer, because it depends on how "nation" and "continuous" are defined. Japan is most often named as the oldest continuous monarchy and sovereign state (traditionally 660 BCE, historically firmer from about the 5th to 7th century CE). San Marino (traditionally 301 CE) is often named as the oldest surviving republic.
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The answer depends on the definition. If you count continuous civilization or culture, Egypt, China, Iran and others are older. If you count an unbroken sovereign state with continuous identity and government, the usual candidates are San Marino (about 301 CE, by tradition) and Japan (a continuous monarchy, with a disputed early history). Many historians treat San Marino as the oldest extant sovereign state, and Japan as the oldest continuous hereditary monarchy. Foreign occupation, conquest and regime changes weaken the claims of the older civilizations. Japan's occupation after 1945 was a brief interruption, but it didn't end the state.
- Japan: the imperial line is traditionally dated to 660 BCE, but historians find firm evidence only from about the 5th to 6th century CE.
- San Marino claims founding in 301 CE, which would make it the oldest surviving sovereign state and republic.
- Egypt, Iran (Persia), China and Ethiopia have much older civilizations, but each had major breaks, conquests or changes of regime and state form.
- Modern Egypt, Iran and China as sovereign states in their current form date from the 20th century, or from the 1930s to 1950s, even though their cultures are ancient.
- Armenia, Georgia and Korea also claim very long continuity, with disputed breaks in sovereignty.