Overview
1039 is a natural number that appears immediately after 1038 and before 1040. In arithmetic it is notable for being a prime number: it has no positive divisors other than 1 and itself. As an integer it is odd and arises in ordinary counting, numbering systems, and sometimes as an identifier (model numbers, route numbers, etc.).
Mathematical properties
As a prime, 1039 is indivisible by any smaller positive integer except 1. It is the 175th prime number. Because 1039 ≡ 3 (mod 4), it cannot be expressed as a sum of two integer squares (by Fermat's theorem on sums of two squares). It is not a twin prime (neither 1038 nor 1040 are prime) and it is not a Sophie Germain prime, nor a safe prime.
Numeric representations
- Decimal: 1039
- Binary: 10000001111
- Octal: 2017
- Hexadecimal: 40F
- Sum of decimal digits: 1 + 0 + 3 + 9 = 13 (digital root 4)
Contexts and occurrences
Numbers like 1039 appear in everyday contexts as counts, labels, and identifiers. In mathematics it serves as an example of a mid-size prime with simple congruence properties used when illustrating modular arithmetic or primality tests. In computational settings, its binary and hexadecimal forms (10000001111b, 0x40F) may be used in examples of bit patterns or memory addresses.
Historical note
The numeral 1039 also denotes the year 1039 in the Julian calendar (eleventh century). That year is part of medieval chronology and appears in historical lists and annals, but the number itself is best studied mathematically as an example of a prime with specific modular behaviour.