1267 (number and year)

Mathematical properties

In arithmetic terms, 1267 is a composite, odd, squarefree integer whose prime factorization is 7 × 181. Because it is the product of two primes it is a semiprime. Its positive divisors are 1, 7, 181 and 1267. The sum of proper divisors is 189, which makes 1267 a deficient number (the proper divisors sum to less than the number itself). The Euler totient function φ(1267) = (7−1)(181−1) = 1080. The total sum of divisors σ(1267) = 1456, and the count of positive divisors τ(1267) = 4.

• Binary: 10011110011
• Hexadecimal: 0x4F3
• Roman numerals: MCCLXVII
• Not expressible as a sum of two integer squares (because it has a prime factor congruent to 3 mod 4 with odd exponent).

Context and historical note for the year 1267

The year 1267 lies within the High Middle Ages, an era of demographic growth, urbanization and institutional development across Europe. It is also part of the century when the Mongol Empire reshaped political landscapes across Eurasia and when long-distance trade and cultural exchange increased. Intellectual life in Europe included scholastic theology and the growth of universities; figures such as Thomas Aquinas were active in this general period. Local and regional political events—dynastic struggles, territorial treaties and ecclesiastical decisions—shaped the year in different places, though specifics vary by region.

Because 1267 is a concise semiprime example, it is sometimes used in elementary number-theory illustrations and exercises. Semiprimes overall have modern relevance as the mathematical foundation for public-key cryptography (large semiprimes are used in schemes such as RSA), though 1267 itself is far too small for practical security.

In summary, 1267 is an instructive integer: small enough to compute basic functions by hand but representative of classes (odd, semiprime, squarefree, deficient) that are meaningful in number theory. As a calendar year, 1267 sits in a dynamic period of medieval history with developments across politics, religion and learning.