17 (number)

Seventeen is the integer after 16 and before 18. It is an odd prime with special algebraic and geometric properties, notable cultural associations, and simple numeric representations like XVII, 10001₂, and 11₁₆.

Overview

Seventeen (written 17) is the natural number following sixteen and preceding eighteen. It is an odd integer and the seventh prime number, coming after 2, 3, 5, 7, 11 and 13. The numeral is commonly written in Roman numerals as XVII. For a general reference to the number in language and notation see seventeen.

Mathematical properties

Seventeen has several notable mathematical features. It is prime, so its only positive divisors are 1 and itself. It forms a twin-prime pair with 19, and it can be expressed as the sum of the first four prime numbers: 2 + 3 + 5 + 7 = 17. As a prime congruent to 1 modulo 4, 17 is representable as a sum of two squares (1² + 4² = 17). It is also a Fermat prime (one of the special primes of the form 2^{2^n} + 1).

Algebraic and geometric significance

Because 17 is a Fermat prime, it has geometric consequences: a regular 17-sided polygon (heptadecagon) is constructible with straightedge and compass. This was proved by Carl Friedrich Gauss and was historically important in the development of classical geometric construction theory. For basic discussion of primality and related sequences consult sources on prime numbers.

Representations and notation

Seventeen admits simple representations in common numeral systems: in binary it is written as 10001, and in hexadecimal as 11. Its Roman numeral representation is XVII; for general rules and history of Roman numerals see Roman numerals. In many contexts the symbol 17 and the word seventeen are used interchangeably to label items, ages, or positions in ordered lists.

Cultural and miscellaneous notes

Different cultures attach various meanings to the number 17. In Italy it is often regarded as unlucky; traditional explanations point to linguistic or historical associations. Elsewhere the number carries neutral or context-dependent significance, such as an age milestone in some legal systems or a motif in literature and music. For cultural references and numerological treatments see general cultural number studies at odd number resources.

Quick facts

Position: 17 is the 7th prime number.
Parity: odd.
Binary: 10001₂; Hexadecimal: 11₁₆.
Special types: Fermat prime; sum of two squares; constructible polygon (regular 17-gon).