821

821 is a natural number that follows 820 and precedes 822. In mathematics, it is notable for being a prime number, since its only positive divisors are 1 and 821. It is not divisible by smaller primes such as 2, 3, 5, 7, 11, 13, 17, 19, or 23, and its square root is about 28.7, which confirms its primality without the need to test larger factors. The digit sum of 821 is 11, and it appears in various integer sequences as a representative prime.

In other numeral systems, 821 has distinct representations: in binary it is 1100110101, in octal 1465, and in hexadecimal 0x335. In Roman numerals it is written as DCCCXXI.

As a calendar year, 821 CE belongs to the early medieval period. It marks a point in the histories of various regions, including the Carolingian Empire and the broader Eurasian worlds of the time. Beyond mathematics and chronology, 821 also appears as a commonly used identifier in cataloging, product models, or organizational codes, where the number functions independently of its numerical properties.

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