1139 is the integer after 1138 and before 1140. In mathematics it is most notable as a composite odd number whose prime factorization is the product of two distinct primes, 17 and 67. This makes 1139 a semiprime and a squarefree integer with exactly four positive divisors.
Mathematical properties
Key numerical facts about 1139 include its divisors and arithmetic functions. Its positive divisors are 1, 17, 67 and 1139. The sum of divisors σ(1139) equals 1 + 17 + 67 + 1139 = 1224, so the sum of proper divisors is 85, which is less than 1139; hence 1139 is a deficient number. The Euler totient function φ(1139) = (17−1)(67−1) = 16 × 66 = 1056. The Carmichael function λ(1139), the exponent of the multiplicative group modulo 1139, is lcm(16,66) = 528.
Representations
- Prime factorization: 1139 = 17 × 67
- Binary (base 2): 10001110011
- Octal (base 8): 2163
- Hexadecimal (base 16): 0x473
- Roman numerals: MCXXXIX
Context and historical year
When read as a calendar year, AD 1139 marks several events of medieval European history. The Second Lateran Council convened in 1139 and addressed issues of ecclesiastical discipline and clerical reform. Around the same period, in the Iberian Peninsula, actions attributed to Afonso Henriques culminated in assertions of independent rulership that later fed into the foundation of the Kingdom of Portugal. These years are commonly cited in surveys of 12th-century church councils and the consolidation of emerging medieval states.
Uses and notable distinctions
As with many integers in this range, 1139 appears in numbered lists, catalogues, and designations—such as document numbers, model identifiers, and route numbers—without carrying special universal significance beyond its arithmetical characteristics. Mathematically, its status as a semiprime and squarefree composite gives it a simple structure that makes it useful in elementary examples of multiplicative functions and modular arithmetic.
Overall, 1139 is a straightforward example in number theory of a two-factor composite with readily computed totient and divisor sums, and as a year it falls within a period of notable ecclesiastical and political developments in medieval Europe.