Overview
1284 is both an integer with several elementary arithmetic properties and a calendar year in the late thirteenth century. As a number it is even, composite and abundant; as a year it is associated with events that affected maritime power in Italy and the political order of parts of the British Isles and Iberia.
Mathematical properties
The prime factorization of 1284 is 2^2 · 3 · 107. Its positive divisors are 1, 2, 3, 4, 6, 12, 107, 214, 321, 428, 642 and 1284, a total of 12 divisors. The sum of all divisors is 3024, so the sum of proper divisors is 1740 and 1284 is classified as an abundant number (abundance 456). The divisor-count function τ(1284)=12 and since 12 divides 1284, the number is an example of a refactorable (tau) number.
Other useful arithmetical data: Euler's totient φ(1284)=424, the binary representation is 10100000100 and the hexadecimal form is 0x504. Because the odd prime factors 3 and 107 are congruent to 3 mod 4 and occur to the first power, 1284 cannot be expressed as a sum of two integer squares. These basic invariants place 1284 among familiar composite integers used in elementary number theory and applications.
Year AD 1284 — notable events
The year 1284 saw several events of lasting regional consequence. In the Mediterranean, the naval Battle of Meloria (1284) resulted in a decisive victory for the Republic of Genoa over the Republic of Pisa; the defeat severely weakened Pisa's maritime position and contributed to Genoa's rise in Mediterranean trade and influence. In the British Isles, 1284 is noted for measures that consolidated English control in Wales following Edward I's campaigns, commonly associated with the Statute of Rhuddlan which established new administrative arrangements for the conquered territories. In the Iberian Peninsula the death of King Alfonso X of Castile in 1284 altered the line of succession and helped shape subsequent Castilian politics and alliances.
Legacy and context
- Numeric: 1284 appears in combinatorial counts and computing contexts where its divisibility and representations are convenient; its totient and divisor structure make it a concrete example in elementary number-theory problems.
- Historical: events of 1284 illustrate the shifting balance of naval power in the Mediterranean and the extension of centralized authority in parts of medieval Europe during the High Middle Ages.
Whether treated as an abstract integer or as a point on a historical timeline, 1284 connects concise mathematical properties with concrete human events, showing how a single label can have distinct but complementary meanings in different disciplines.