Overview

841 is a natural number that lies between 840 and 842. As an integer it is notable for being a perfect square: 841 = 29². That simple identity shapes many of its elementary arithmetic properties. The same numeral also designates the year 841 in the common era, a date situated in the ninth-century Carolingian period of European history.

Mathematical properties

Because 841 = 29² and 29 is prime, 841 has a very small divisor set. Its positive divisors are 1, 29 and 841. As a square of a prime, its number-of-divisors function τ(841) = 3 and its Euler totient φ(841) = 29·28 = 812. The sum of its positive divisors σ(841) = 1 + 29 + 841 = 871. Being a prime square, 841 is neither prime nor squarefree.

Representations and residues

In other numeral systems 841 appears as 1101001001 in binary, 1511 in octal and 0x349 in hexadecimal. Its decimal digit sum is 8 + 4 + 1 = 13, giving a digital root of 4. Modulo arithmetic yields 841 ≡ 1 (mod 4) and 841 ≡ 1 (mod 3), reflecting properties inherited from its prime base 29.

Context and examples

As 29², 841 is the sum of the first 29 odd numbers (a standard characterization of square numbers). It sits between 28² = 784 and 30² = 900, and therefore often appears in elementary textbook examples about squares, factors, and arithmetic functions. Its compact factorization makes it straightforward to use in problems illustrating divisor sums, Möbius and totient computations.

Year 841 (AD)

The year 841 falls in the later Carolingian period. It is remembered in broad terms for military and political struggles among the heirs of Louis the Pious as the dynasty fragmented after his death. One significant engagement of the era was the clash between rival Carolingian princes that helped determine the future partitioning of West and East Francia.

Notable facts and distinctions

  • 841 is a perfect square (29²) and therefore has an odd number of divisors.
  • Its simplicity (prime-square) makes it a common example in elementary number theory.
  • As a historical year it illustrates the broader pattern of ninth-century dynastic conflict in western Europe rather than a single defining event.