1081 can refer either to the integer that follows 1080 or to the year 1081 in the Common Era. Both senses appear in mathematics and history: as a number it has simple but notable factorization and arithmetic features; as a year it belongs to a period of political change in the Mediterranean and Europe.
Mathematical properties
The integer 1081 is composite and specifically a semiprime: its prime factorization is 23 × 47. Both factors are prime, and 23 is a Sophie Germain prime because 2×23+1 = 47 is also prime; thus 1081 is the product of a Sophie Germain prime and its associated safe prime. Its positive divisors are 1, 23, 47 and 1081. Arithmetic functions give φ(1081) = (23−1)(47−1) = 1,012, and the sum of all divisors is 1,152, so the sum of proper divisors (71) is less than 1081: it is a deficient number. In Roman numerals 1081 is written as MLXXXI.
Year 1081 (AD)
The year 1081 lies in the late 11th century, a time of shifting power in southern Europe and the eastern Mediterranean. In that year Alexios I Komnenos became emperor of the Byzantine Empire, beginning a dynasty that sought to restore imperial strength after years of internal strife and external pressure. The same period saw Norman leaders from southern Italy pressing into Byzantine-held territories; a major confrontation between Norman forces under Robert Guiscard and Byzantine troops occurred near the Adriatic coast, affecting control of parts of Greece and the western Balkans.
Uses, examples and significance
As a number, 1081 appears in elementary number theory examples illustrating semiprimes, multiplicative functions, and factorization exercises. As a historical year, 1081 is often cited in overviews of the Komnenian restoration of Byzantium and in studies of Norman expansion in Italy and the Balkans. The political developments of 1081 contributed to the broader context in which Western and Eastern Christendom interacted in the decades that followed.
Quick facts
- Integer type: odd semiprime (23 × 47).
- Divisors: 1, 23, 47, 1081; proper divisor sum = 71 (deficient).
- Euler's totient φ(1081) = 1,012.
- Historical highlight (AD 1081): accession of Alexios I Komnenos and Norman–Byzantine conflicts.