1428 (number and year)

Overview of 1428 as an integer (arithmetic properties, factorization and representations) and as the calendar year MCDXXVIII, with historical context and notable facts.

Overview

1428 is both a natural number that appears in elementary number theory and the designation of the year MCDXXVIII in the late Middle Ages. As an integer it lies between 1427 and 1429 and exhibits several standard arithmetic properties used to illustrate multiplicative functions, divisor sums and basic factorization. As a year it belongs to the 15th century and is associated with events in the Hundred Years' War and the wider political landscape of medieval Europe.

Mathematical properties

In prime factorization 1428 = 2^2 × 3 × 7 × 17. From this factorization follow several commonly cited quantities:

Number of positive divisors: (2+1)(1+1)(1+1)(1+1) = 24.
Sum of all positive divisors σ(1428) = 4032, so the sum of proper divisors is 4032 − 1428 = 2604, making 1428 an abundant number (proper divisors sum exceeds the number).
Euler's totient φ(1428) = 1428·(1−1/2)(1−1/3)(1−1/7)(1−1/17) = 384, the count of integers ≤1428 coprime to it.

Other standard representations: Roman numeral MCDXXVIII, binary 10110010100, and hexadecimal 594 (base-16). It is even, composite, and not a perfect number, not a prime power, and — because it has prime factors congruent to 3 mod 4 with odd exponents — it is not expressible as a sum of two integer squares.

Year 1428 (MCDXXVIII)

The year 1428 in the Julian calendar was a leap year. One of the principal historical events that began in 1428 was the Siege of Orléans, which opened in the autumn and became a pivotal episode in the later stages of the Hundred Years' War between England and France. The siege set the stage for a series of military and political developments in 1429 that altered the course of the conflict.

More generally, 1428 falls within a period of regional warfare, dynastic competition and gradual social change across Europe and beyond. Local chronicles and administrative records from the early 15th century commonly record military campaigns, transfers of titles and routine ecclesiastical business; these provide the primary evidence for reconstructing events of the year.

Uses and context

The integer 1428 is used today as an identifier in addresses, catalogue numbers and models, and serves in pedagogy as a convenient example for explaining divisor functions and multiplicative arithmetic functions. The year 1428 is most often referenced in historical summaries for its connection to the Siege of Orléans and the broader narrative of the Hundred Years' War, while many other local or regional occurrences of the year are known from specialized archival sources.