17 (number)

Last updated
16 17 18
Cardinal seventeen
Ordinal 17th
(seventeenth)
Numeral system septendecimal
Factorization prime
Prime 7th
Divisors 1, 17
Greek numeral ΙΖ´
Roman numeral XVII
Binary 100012
Ternary 1223
Senary 256
Octal 218
Duodecimal 1512
Hexadecimal 1116
Hebrew numeral י"ז
Babylonian numeral 𒌋𒐛

17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.

Contents

Seventeen is the sum of the first four prime numbers.

17 was described at MIT as "the least random number", according to the Jargon File. [1] 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. [2]

Mathematics

Seventeen is the seventh prime number, which makes it the fourth super-prime, [3] as seven is itself prime.

Prime properties

Seventeen is the only prime number which is the sum of four consecutive primes (2, 3, 5, and 7), as any other four consecutive primes that are added always generate an even number divisible by two.

It forms a twin prime with 19, [4] a cousin prime with 13, [5] and a sexy prime with both 11 and 23. [6] Furthermore,

The number of integer partitions of 17 into prime parts is 17 (the only number such that its number of such partitions is ). [11]

Fermat prime

Seventeen is the third Fermat prime, as it is of the form with . [12] On the other hand, the seventeenth Jacobsthal–Lucas number — that is part of a sequence which includes four Fermat primes (except for 3) — is the fifth and largest known Fermat prime: 65,537. [13] It is one more than the smallest number with exactly seventeen divisors, 65,536 = 216. [14]

Since seventeen is a Fermat prime, regular heptadecagons can be constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies. [15] [16]

Quadratic integer matrix

A positive definite quadratic integer matrix represents all primes when it contains at least the set of seventeen numbers:

Only four prime numbers less than the largest member are not part of the set (53, 59, 61, and 71). [17]

Geometric properties

Two-dimensions

The Spiral of Theodorus, with a maximum sixteen right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a hypotenuse of
17
.
{\displaystyle {\sqrt {17}}.} Wheel of Theodorus.png
The Spiral of Theodorus, with a maximum sixteen right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a hypotenuse of
  • Either 16 or 18 unit squares can be formed into rectangles with perimeter equal to the area; and there are no other natural numbers with this property. The Platonists regarded this as a sign of their peculiar propriety; and Plutarch notes it when writing that the Pythagoreans "utterly abominate" 17, which "bars them off from each other and disjoins them". [28]

17 is the least for the Theodorus Spiral to complete one revolution. [29] This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with . 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 integers from 3 to 17 are irrational by means of this spiral.

Enumeration of icosahedron stellations

In three-dimensional space, there are seventeen distinct fully supported stellations generated by an icosahedron. [30] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules. [31] [32] 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). [33] Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron. [30]

Four-dimensional zonotopes

Seventeen is also the number of four-dimensional parallelotopes that are zonotopes. Another 34, or twice 17, are Minkowski sums of zonotopes with the 24-cell, itself the simplest parallelotope that is not a zonotope. [34]

Abstract algebra

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank mirror facets, with the lowest belonging to the third. [35]

17 is a supersingular prime, because it divides the order of the Monster group. [36] If the Tits group is included as a non-strict group of Lie type, then there are seventeen total classes of Lie groups that are simultaneously finite and simple (see classification of finite simple groups). In base ten, (17, 71) form the seventh permutation class of permutable primes. [37]

Other notable properties

Complex analysis

There are seventeen orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the three-variable Laplace equation can be solved using the separation of variables technique.

Sudoku puzzle

The minimum possible number of givens for a sudoku puzzle with a unique solution is 17. [39] [40]

In science

The elementary particles in the Standard Model of physics Standard Model of Elementary Particles.svg
The elementary particles in the Standard Model of physics

Physics

Seventeen is the number of elementary particles with unique names in the Standard Model of physics. [41]

Chemistry

Group 17 of the periodic table is called the halogens. The atomic number of chlorine is 17.

Biology

Some species of cicadas have a life cycle of 17 years (i.e. they are buried in the ground for 17 years between every mating season).

In religion

Other fields

Seventeen is:

Music

Where Pythagoreans saw 17 in between 16 from its Epogdoon of 18 in distaste, [42] the ratio 18:17 was a popular approximation for the equal tempered semitone (12-tone) during the Renaissance.

Notes

    Related Research Articles

    2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and the only even prime number.

    15 (fifteen) is the natural number following 14 and preceding 16.

    90 (ninety) is the natural number following 89 and preceding 91.

    23 (twenty-three) is the natural number following 22 and preceding 24.

    73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.

    31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

    58 (fifty-eight) is the natural number following 57 and preceding 59.

    63 (sixty-three) is the natural number following 62 and preceding 64.

    1000 or one thousand is the natural number following 1M

    300 is the natural number following 299 and preceding 301.

    500 is the natural number following 499 and preceding 501.

    700 is the natural number following 699 and preceding 701.

    600 is the natural number following 599 and preceding 601.

    <span class="mw-page-title-main">1,000,000,000</span> Natural number

    1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. With a number, "billion" can be abbreviated as b, bil or bn.

    138 is the natural number following 137 and preceding 139.

    257 is the natural number following 256 and preceding 258.

    177 is the natural number following 176 and preceding 178.

    288 is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".

    20,000 is the natural number that comes after 19,999 and before 20,001.

    60,000 is the natural number that comes after 59,999 and before 60,001. It is a round number. It is the value of (75025).

    References

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