Overview
1331 is a positive integer notable for several elementary number-theoretic and representational properties. It equals 11³, making it a perfect cube and a prime power. Written in base 10 it reads the same forwards and backwards, so it is also a palindromic number.
Basic characteristics
As the cube of a prime, 1331 has a simple prime factorization: 1331 = 11³. Its positive divisors are 1, 11, 121 and 1331, so it has exactly four divisors. The sum of its decimal digits is 1 + 3 + 3 + 1 = 8, and in Roman numerals it is MCCCXXXI.
Representations and patterns
Several curious representations highlight arithmetic patterns. Because 1331 = 11³ = (10 + 1)³, its decimal digits 1–3–3–1 match the binomial coefficients of (x + y)³ before any carry operations, a manifestation of how small powers of 11 reproduce rows of Pascal's triangle (for example, 11² = 121, 11³ = 1331, 11⁴ = 14641). In base 11, 1331 is written as "1000", since it is exactly 11³. Its binary form is 10100110011.
Algebraic and arithmetic notes
Because 1331 is p³ for the prime p = 11, it is a cube of a prime and thus a perfect power. Classical results about sums of two squares imply that 1331 cannot be represented as a sum of two integer squares: the prime factor 11 is congruent to 3 mod 4 and appears to an odd exponent. As a consequence of being a prime power, many multiplicative functions take simple values on 1331 (for example, the number of divisors is 4).
Contexts and curiosities
Outside abstract arithmetic, 1331 appears whenever powers of 11 are considered, in recreational-number puzzles, and as a calendar year label (1331 AD/CE) in historical chronology. It shares the status of palindromic cube with 343 (7³) and other small cubes that form palindromes in particular bases. Such coincidences are used frequently to illustrate elementary combinatorial and positional-number ideas.
- Prime factorization: 11³
- Divisors: 1, 11, 121, 1331
- Decimal palindrome: 1331
- Base‑11 notation: 1000₁₁