17 (seventeen) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 16 and preceding 18. It is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

. 17 was described at

MIT The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...

as "the least random number", according to the

Jargon File The Jargon File is a glossary and usage dictionary of slang used by computer programmers. The original Jargon File was a collection of terms from technical cultures such as the MIT Computer Science and Artificial Intelligence Laboratory, MIT AI Lab ...

. This is supposedly because, in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice. This study has been repeated a number of times.

Mathematics

17 is a

Leyland number In number theory, a Leyland number is a number of the form :x^y + y^x where ''x'' and ''y'' are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are : 8, 17, 32, 54, 57, 100, 145, ...

and Leyland prime, using 2 & 3 (2³ + 3²) and using 4 and 5, using 3 & 4 (3⁴ - 4³). 17 is a

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...

. 17 is one of six lucky numbers of Euler. Since seventeen is a Fermat prime, regular

heptadecagon In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. Regular heptadecagon A ''regular polygon, regular heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular he ...

s can be constructed with a

compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...

and unmarked ruler. This was proven by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

and ultimately led him to choose mathematics over philology for his studies. The minimum possible number of givens for a

sudoku Sudoku (; ; originally called Number Place) is a logic puzzle, logic-based, combinatorics, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row, and ...

puzzle with a unique solution is 17.

Geometric properties

Two-dimensions

*There are seventeen crystallographic space groups in two dimensions. These are sometimes called

wallpaper group A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...

s, as they represent the seventeen possible symmetry types that can be used for

wallpaper Wallpaper is used in interior decoration to cover the interior walls of domestic and public buildings. It is usually sold in rolls and is applied onto a wall using wallpaper paste. Wallpapers can come plain as "lining paper" to help cover uneve ...

. *Also in two dimensions, seventeen is the number of combinations of regular polygons that completely fill a plane vertex. Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, and 5.5.10) exclusively surround a point in the plane and fill it only when irregular polygons are included. *Seventeen is the minimum number of vertices on a two-dimensional

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

such that, if the edges are colored with three different colors, there is bound to be a

monochromatic triangle In graph theory and theoretical computer science, the monochromatic triangle problem is an algorithmic problem on graphs, in which the goal is to partition the edges of a given graph into two triangle-free subgraphs. It is NP-complete but fixe ...

; see

Ramsey's theorem In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (sa ...

. *Either 16 or 18

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...

s can be formed into rectangles with perimeter equal to the area; and there are no other

s with this property. The

Platonist Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...

s regarded this as a sign of their peculiar propriety; and

Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...

notes it when writing that the

Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...

"utterly abominate" 17, which "bars them off from each other and disjoins them". 17 is the least

k

for the Theodorus Spiral to complete one

revolution In political science, a revolution (, 'a turn around') is a rapid, fundamental transformation of a society's class, state, ethnic or religious structures. According to sociologist Jack Goldstone, all revolutions contain "a common set of elements ...

. This, in the sense of

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

, who questioned why Theodorus (his tutor) stopped at

\sqrt

when illustrating adjacent

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

s whose bases are

unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...

s and heights are successive

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

s, starting with

1

. In part due to Theodorus’s work as outlined in Plato’s '' Theaetetus'', it is believed that Theodorus had proved all the square roots of non-

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

integers from 3 to 17 are

irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...

by means of this spiral.

Enumeration of icosahedron stellations

In three-dimensional space, there are seventeen distinct fully supported stellations generated by an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

. The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules. Without counting the icosahedron as a ''zeroth'' stellation, this total becomes 58, a count equal to the sum of the first seven prime numbers (2 + 3 + 5 + 7 ... + 17). Seventeen distinct fully supported stellations are also produced by

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangle (geometry), triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triak ...

and

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

Four-dimensional zonotopes

Seventeen is also the number of four-dimensional parallelotopes that are zonotopes. Another 34, or twice 17, are

Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

s of zonotopes with the

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

, itself the simplest parallelotope that is not a zonotope.

Abstract algebra

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank

n+2

mirror facets, with the lowest belonging to the third. 17 is a supersingular prime, because it divides the order of the

Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 ...

. If the

Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 17,971,200 = 211 · 33 · 52 · 13. This is the only simple group that is a derivativ ...

is included as a ''non-strict'' group of Lie type, then there are seventeen total classes of

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s that are simultaneously

finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

and

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

(see

classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...

). In

base ten The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of t ...

, (17, 71) form the seventh permutation class of

permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given radix, base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to stu ...

Other notable properties

* The sequence of residues (mod ) of a

googol A googol is the large number 10100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, ...

and

googolplex A googolplex is the large number 10, or equivalently, 10 or . Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes; that is, a 1 followed by a googol of zeroes. Its prime factorization is 2 ×5. History In 1920, ...

, for

n=1, 2, 3, ...

, agree up until

n=17

. * Seventeen is the longest sequence for which a solution exists in the

irregularity of distributions The irregularity of distributions problem, stated first by Hugo Steinhaus, is a numerical problem with a surprising result. The problem is to find ''N'' numbers, x_1,\ldots,x_N, all between 0 and 1, for which the following conditions hold: * The f ...

problem.

Other fields

Music

Where

saw 17 in between 16 from its

Epogdoon In Western culture, Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a interval (music), musical interval encompassing two adjacent staff positions ( ...

of 18 in distaste, the ratio 18:17 was a popular approximation for the

equal tempered An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system ...

semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...

(12-tone) during the

Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...

Notes

References

External links

Prime Curios!: 17

Is 17 the "most random" number?
{{DEFAULTSORT:17 (Number) Integers