1159

Mathematical properties

As an integer, 1159 is an odd composite number. Its prime factorization is 19 × 61, so it is a semiprime (the product of two prime numbers). The positive divisors of 1159 are 1, 19, 61 and 1159. Adding those divisors yields 1,240, so the sum of proper divisors is 81, which is less than 1159; therefore 1159 is a deficient number.

• Prime factors: 19 and 61
• Divisors: 1, 19, 61, 1159
• Sum of proper divisors: 81 (deficient)
• Binary representation: 10010000111
• Hexadecimal representation: 0x487

Because it is the product of two distinct primes, 1159 is of the simple semiprime class and does not meet special criteria like being a square or a prime power. It is not a Blum integer (both prime factors would need to be congruent to 3 mod 4), since 61 ≡ 1 (mod 4).

Year 1159 (AD)

The year 1159 falls in the high Middle Ages, a period marked by dynastic rivalry, church–state tensions, and territorial consolidation across Europe and the Mediterranean. One of the most historically significant occurrences in that year was the papal succession that followed the death of the reigning pope. The college of cardinals produced a contested outcome: a majority elected Pope Alexander III, while an imperial faction supported an alternative claimant, often described as an antipope. That division inaugurated a prolonged schism, reflecting and intensifying the broader conflict between the papacy and imperial authority.

In wider context, the mid-12th century saw rulers such as the Holy Roman Emperor and kings of France and England engaged in shifting alliances and intermittent warfare, while political authority in Italy and the Norman kingdoms of southern Italy and Sicily remained important factors in papal and imperial policy. The schism of 1159 is notable for illustrating how ecclesiastical elections could become focal points for secular power struggles.

Uses, references and notable facts

The number 1159 appears in a variety of practical contexts as an identifier: catalog numbers, model names, route numbers, and dates. Mathematically, it serves as a straightforward example when illustrating semiprimes, factorization, and divisor functions in elementary number theory. Historically, references to "1159" without further context most often concern the papal election and the ensuing contest for legitimacy among competing ecclesiastical and imperial factions.