1333 — the integer and the year AD 1333

Mathematical properties

Arithmetically, 1333 is a composite integer and specifically a semiprime: it factors as 31 × 43. Its positive divisors are 1, 31, 43 and 1333. The sum of all divisors is 1,408, so the sum of proper divisors is 75, which is less than 1333 and therefore the number is classified as deficient. Euler's totient function for 1333 is φ(1333) = (31 − 1)(43 − 1) = 1,260.

• Prime factorization: 1333 = 31 × 43
• Binary: 10100110101; octal: 2465; hexadecimal: 0x535
• Arithmetic types: semiprime and Blum integer (both prime factors ≡ 3 mod 4)

Being a Blum integer (both prime factors congruent to 3 modulo 4) places 1333 in a class of numbers that are studied in number theory for properties of quadratic residues and in illustrative examples related to cryptographic constructions; practical cryptography uses much larger integers, but small Blum integers serve as accessible examples in proofs and teaching.

Year AD 1333: historical context

The year AD 1333 was notable in several regions. In Japan it marked the collapse of the Kamakura shogunate: a series of military actions culminated in the fall of Kamakura and a brief restoration of imperial authority under Emperor Go‑Daigo in the episode known as the Kemmu Restoration. Key figures in the conflict included samurai leaders who contested shogunal rule and sought realignment of political authority.

In the British Isles, 1333 saw the Battle of Halidon Hill (19 July 1333), when forces of King Edward III of England defeated a Scottish army in a decisive engagement during the Second War of Scottish Independence; the battle affected the siege and control of the border town of Berwick‑upon‑Tweed and had significant military and political consequences for Anglo‑Scottish relations in that period.

These events form part of broader 14th‑century dynamics characterized by shifting military power, contested dynastic claims and regional reorganisation. For earlier eras, the label "1333 BC" is a chronological marker placed within the Late Bronze Age; synchronisms for that distant period remain subject to scholarly debate and refinement.

Other references

As with many three‑ or four‑digit numbers, 1333 is used in catalogues, model numbers, route designations and identifiers in various administrative or technical systems; such usages are arbitrary and do not reflect intrinsic mathematical properties. In academic and recreational contexts the number is most often discussed for its elementary number‑theoretic features and for the historical events attached to the year AD 1333.