1146 (AD year and integer)

Year 1146 — overview and context

In 1146 Bernard of Clairvaux, the influential Cistercian abbot, preached widely in France and Germany to promote a new crusading effort after the loss of Edessa in 1144. His sermons and personal influence contributed to leaders such as Louis VII of France and Conrad III of Germany committing to the cause. Key gatherings and public assemblies in 1146 saw many nobles and knights publicly undertake vows that culminated in organized expeditions departing in 1147. The broad religious and political atmosphere of the period included monastic expansion, ongoing ecclesiastical reform, and growing centralized royal authority in parts of western Europe.

Notable developments and broader setting

1146 sits within the era of the Komnenian Byzantine Empire under Manuel I and within the longer processes of the Reconquista in Iberia and of territorial consolidation in western kingdoms. Monastic orders such as the Cistercians expanded across Europe, influencing economic and spiritual life. The mobilization and rhetoric of 1146 helped transform scattered appeals into a general movement that produced one of the major crusading expeditions of the twelfth century.

1146 as an integer — arithmetic characteristics

• Factorization and primes: 1146 = 2 × 3 × 191, so it is the product of three distinct prime factors.
• Sphenic: because it is the product of three different primes, 1146 is classified as a sphenic number.
• Divisors and abundance: its positive divisors are 1, 2, 3, 6, 191, 382, 573 and 1146. The sum of proper divisors is 1+2+3+6+191+382+573 = 1158, which exceeds 1146 by 12, so 1146 is an abundant number with abundance 12.
• Arithmetic functions: the sum of all divisors (sigma) is 2304, the number of divisors (tau) is 8, and Euler's totient function yields phi(1146) = 380.
• Other forms: in Roman numerals 1146 is written MCXLVI; it is an even composite integer and sometimes appears in routine contexts such as indexing, cataloguing or simple numerical examples.

Significance and usage

As a historical label, 1146 is a convenient chronological marker when discussing the buildup and public rhetoric that produced the Second Crusade. It is referenced in studies of twelfth-century monasticism and ecclesiastical influence on politics. As a number, 1146 serves as a clear example in teaching factorization, divisor sums and multiplicative arithmetic functions. Whether treated as a year or as a number, 1146 occupies familiar ground in both medieval history and elementary number theory.