1073 (number and year)

Mathematical properties

In prime factorization 1073 = 29 × 37, so it is a semiprime (the product of two prime numbers) and squarefree. Its positive divisors are 1, 29, 37 and 1073. The sum of all divisors (sigma function) is 1 + 29 + 37 + 1073 = 1140, and the sum of proper divisors is 67, which is less than 1073, so 1073 is a deficient number. Euler's totient function for 1073 equals (29−1)(37−1) = 28 × 36 = 1008.

Representations and patterns

1073 has simple representations in several bases: in binary it is 10000110001, in octal 2061, and in hexadecimal 0x431. It can be expressed as a sum of two squares: 1073 = 32² + 7², a consequence of its prime factors both being congruent to 1 modulo 4. Because it is a product of two distinct primes, it is an example often used in elementary demonstrations about semiprimes and basic multiplicative functions.

Uses and examples

While 1073 is far too small to have cryptographic significance, semiprimes like it illustrate the factorization concepts that underpin public-key cryptosystems. In recreational mathematics it can serve as a simple test case for algorithms that compute divisors, totients, and representations as sums of squares.

Year AD 1073: brief historical note

In the historical year 1073 (AD), a notable event was the election of Hildebrand of Sovana as pope, who took the name Gregory VII; his pontificate would play a major role in the Investiture Controversy later in the 11th century. Other regional political and cultural developments of the High Middle Ages continued across Europe, the Byzantine world and the Islamic world during this period.

Notable distinctions

• Arithmetic class: semiprime and squarefree.
• Sum of two squares: 1073 = 32² + 7².
• Deficient number with totient φ(1073) = 1008 and divisor sum σ(1073) = 1140.