In mathematics
Five is the third
prime number, after
2 and
3, and before
7. Because it can be written as 2^(2^1)+1, five is classified as a
Fermat prime. 5 is the third
Sophie Germain prime, the first
safe prime, and the third
Mersenne prime exponent. Five is the first
Wilson prime and the third
factorial prime, also an
alternating factorial. It is an
Eisenstein prime with no imaginary part and real part of the form
. It is also the only number that is part of more than one pair of
twin primes.
Five is conjectured to be the only odd
untouchable number.
The number 5 is the 5th
Fibonacci number, being
2 plus
3. 5 is also a
Pell number and a
Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth
Perrin numbers.
5 and 6 form a
Ruth-Aaron pair under either definition.
There are five solutions to
Znám's problem of length 6.
Five is the second
Sierpinski number of the first kind, and can be written as S2=(2^2)+1
While
polynomial equations of degree
4 and below can be solved with radicals, equations of degree 5 and higher cannot generally be so solved. This is the
Abel-Ruffini theorem. This is related to the fact that the
symmetric group ''S''
''n'' is a
solvable group for ''n'' ≤ 4 and not solvable for ''n'' ≥ 5.
While all
graphs with 4 or fewer vertices are
planar, there exists a graph with 5 vertices which is not planar: ''K''
5, the
complete graph with 5 vertices.
Five is also the number of
Platonic solids.
A
polygon with five sides is a
pentagon.
Figurate numbers representing pentagons (including five) are called
pentagonal numbers. Five is also a
square pyramidal number.
Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this, 5 is in base 10 a 1-
automorphic number.
Vulgar fractions with 5 or
2 in the
denominator do not yield infinite
decimal expansions, as is the case with most primes, because they are prime factors of
ten, the base. When written in the decimal system, all multiples of 5 will end in either 5 or
0.
There are five
Exceptional Lie groups.
The number of
terminal zeros in any number of numbers multiplied together will typically equal the number of 5's found in the prime factorization of the numbers. This means that multiplying the first 100 integers together will result in a number with 24
terminal zeros
Evolution of the glyph
The evolution of our modern glyph for five cannot be neatly traced back to the Brahmin Indians quite the same way it can for 1 to 4. Later on the Kushana and Gupta Indians had among themselves several different glyphs which bear no resemblance to the modern glyph. The Nagari and Punjabi took these glyphs and all came up with glyphs that look like a lowercase "h" rotated 180°. The Ghubar Arabs transformed the glyph in several different ways, coming up with glyphs that look more like 4s or 3s than 5s.
[Georges Ifrah, ''The Universal History of Numbers: From Prehistory to the Invention of the Computer'' transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.65] It was from those characters that the Europeans finally came up with the modern 5, though from purely graphical evidence, it would be much easier to conclude that our modern 5 came from the
Khmer.
While the shape of the 5 character has an
ascender in most modern
typefaces, in typefaces with
text figures the character usually has a
descender, as, for example, in .