Five

Five is the second Sierpinski number of the first kind, and can be written as S2=(2^2)+1

Five is the third smallest 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 3n - 1. It is also the only number that is part of more than one pair of twin primes.

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.

Five is a factor of 10, so vulgar fractions with 5 in the denominator do not yield infinite decimal expansions, unlike most other primes. When written in the decimal system, all multiples of 5 will end in either 5 or 0.

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), ... (A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5).

There are five solutions to Znám's problem of length 6.

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 Sn 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: K5, 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 answer to the question asked at the very end of the mathematics quiz show in the movie Little Man Tate. (Our young protagonist blurts out the answer, but the host mishears it as being the answer from the contestant to whom the question is posed, and declares him the winner.)

In numbering systems

• In binary code five is 101
• In ternary code five is 12
• In quaternary numeral system code five is 11
• In quinary five is 10; in senary code and all codes above (such as decimal, duodecimal and vigesimal) five is 5.
• The Roman numeral for five is V, which comes from a representation of an outstretched hand.
• In the Greek alphabet, e (epsilon) has numerical value of 5.
• In the Hebrew alphabet, ? (heh) has numerical value of 5.
• In the Cyrillic alphabet, ? has numerical value of 5.
• In the Glagolitic alphabet,  (dobro) has numerical value of 5.
• The kanji and Chinese character for five are both ?, and its formal writing in Chinese is ? (pinyin wu).