5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.

Five is the third-smallest prime number,^[1] equal to the sum of the only consecutive positive integers to also be prime numbers (2 + 3). In integer sequences, five is also the second Fermat prime, and the third Mersenne prime exponent, as well as the fourth or fifth Fibonacci number;^[2] 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).^[3]

In geometry, the regular five-sided pentagon is the first regular polygon that does not tile the plane with copies of itself, and it is the largest face that any of the five regular three-dimensional regular Platonic solid can have, as represented in the regular dodecahedron. For curves, a conic is determined using five points in the same way that two points are needed to determine a line.^[4]

In abstract algebra and the classification of finite simple groups, five is the count of exceptional Lie groups as well as the number of Mathieu groups that are sporadic groups. Five is also, more elementarily, the number of properties that are used to distinguish between the four fundamental number systems used in mathematics, which are rooted in the real numbers.

Historically, 5 has garnered attention throughout history in part because distal extremities in humans typically contain five digits.

In the classification of number systems, the real numbers ${\displaystyle \mathbb {R} }$ and its three subsequent Cayley–Dickson constructions of algebras over the field of the real numbers (i.e. the complex numbers ${\displaystyle \mathbb {C} }$, the quaternions ${\displaystyle \mathbb {H} }$, and the octonions ${\displaystyle \mathbb {O} }$) are normed division algebras that hold up to five different principal algebraic properties of interest: whether the algebras are ordered, and whether they hold commutative, associative, alternative, and power-associative multiplicative properties.^[5] Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. In comparison, the sedenions ${\displaystyle \mathbb {S} }$, which represent a fifth algebra in this series, is not a composition algebra unlike ${\displaystyle \mathbb {H} }$ and ${\displaystyle \mathbb {O} }$, is only power-associative, and is the first algebra to contain non-trivial zero divisors as with all further algebras over larger fields.^[6] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.

Five is the third prime number, and more specifically, the second super-prime since its prime index is prime.^[1] Aside from being the sum of the only consecutive positive integers to also be prime numbers, 2 + 3, it is also the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7);^[7][8] this makes it the first balanced prime with equal-sized prime gaps above and below it (of 2).^[9] 5 is the first safe prime^[10] where ${\displaystyle (p-1)/2}$ for a prime ${\displaystyle p}$ is also prime (2), and the first good prime, since it is the first prime number whose square (25) is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes (i.e., 3 × 7 = 21 and 11 × 2 = 22 are less than 25).^[11] 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5.^[12] More significantly, the fifth Heegner number that forms an imaginary quadratic field with unique factorization is also 11^[13] (and the first repunit prime in decimal, a base in-which five is also the first non-trivial 1-automorphic number).^[14] 5 is also an Eisenstein prime (like 11) with no imaginary part and real part of the form ${\displaystyle 3p-1}$.^[1]

5 is the first prime number (and more generally, natural number) ${\displaystyle n}$ that is palindromic for a base ${\displaystyle b}$ where ${\displaystyle 2\leq b\leq n-2}$, with adjacent numbers 4 and 6 the only two composite numbers to be strictly non-palindromic in such sense.^[15] In other words, all numbers greater than 6 in this sequence are prime, where 11 is the next strictly non-palindromic number after 6, equal to the sum of all non-prime entries in the sequence (0, 1, 4, 6). Positive integers have representations as sums of three palindromic numbers only in bases greater than or equal to five (quinary).^[16]

All prime numbers greater than or equal to 5 are congruent to ${\displaystyle \pm 1{\text{ mod }}6}$ (as well as, ${\displaystyle \pm 1{\text{ mod }}4}$).

5 is the third Mersenne prime exponent ${\displaystyle n}$ for ${\displaystyle 2^n-<!--\; -\; -->1}$Zdroj:https://en.wikipedia.org?pojem=5_(number)
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