Overview

842 is an integer that follows 841 and precedes 843. In arithmetic it appears as a simple even composite number with a small factorization. The symbol 842 also identifies calendar years (842 BC and AD 842) and occasionally appears in catalogues, model numbers and numbered lists. Because of its arithmetic shape and a few historical coincidences, it attracts attention in elementary number theory and in medieval history.

Mathematical properties

As a positive integer, 842 has a compact list of elementary invariants that are often cited in reference works:

  • Prime factorization: 2 × 421, so 842 is a semiprime (a product of two primes) and is even.
  • Divisors: 1, 2, 421, 842. The sum of proper divisors is 424, which is less than 842, so the number is deficient.
  • Euler totient: φ(842) = φ(2)·φ(421) = 1·420 = 420, the count of integers up to 842 that are coprime with it.
  • Representations: binary 1101001010, hexadecimal 34A, and the decimal digit sum 8+4+2 = 14 (digital root 5).
  • Nearest square: 842 = 29² + 1, so it lies one greater than the perfect square 841.

These simple descriptors place 842 among familiar integer classes: it is composite but not highly composite, not perfect, and not a member of common special sets such as triangular or perfect numbers.

Year AD 842

The notation "842" also denotes the year AD 842 in the medieval era. One of the best-known events of that year is the Oaths of Strasbourg, sworn by two of Charlemagne's grandsons. The oaths are historically significant because they survive in two versions that illustrate early vernacular forms of Romance and Germanic speech, and are frequently cited in studies of the development of Old French and Old High German.

Uses and occurrences

Outside mathematics and chronology, the number 842 appears in administrative numbering, product identifiers and catalogue codes. As with many three-digit numbers, it is sometimes chosen for model names or internal identifiers; such usages are arbitrary and do not imply deeper mathematical meaning.

Notable facts and distinctions

Among concise notable points: 842 is an even semiprime, equals one more than the square of 29, and has totient 420. In historical context, AD 842 is remembered for primary source material that illuminates the early medieval linguistic landscape of Western Europe.