5
Integer number 5 / From Wikipedia, the free encyclopedia
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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. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
Five is the third-smallest prime number, and the second super-prime, since its prime index is prime.[1] Notably, 5 is equal to the sum of the only consecutive primes 2 + 3 and it is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7),[2][3] also making it the first balanced prime with equal-sized prime gaps above and below it (of 2).[4] 5 is the first safe prime[5] where for a prime 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).[6] 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5.[7] More significantly, the fifth Heegner number that forms an imaginary quadratic field with unique factorization is also 11[8] (and the first repunit prime in decimal, a base in-which five is also the first non-trivial 1-automorphic number).[9]
- Every odd number greater than is the sum of at most five prime numbers,[10] and every odd number greater than is conjectured to be expressible as the sum of three prime numbers; Helfgott has provided a proof of the latter[11] (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review.
- As a consequence of Fermat's little theorem and Euler's criterion, all squares are congruent to 0, 1, 4 (or −1) modulo 5.[12]
- While polynomial equations of degree 4 and below can be solved with radicals, quintic equations of degree 5 and higher cannot generally be so solved (see, Abel–Ruffini theorem). This is related to the fact that the symmetric group is a solvable group for ⩽ , and not for ⩾ .
- Five is conjectured to be the only odd untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.[13]
Is 5 the only odd untouchable number?
Five is also the second Fermat prime, and the third Mersenne prime exponent, as well as the fourth or fifth Fibonacci number. [14] It is also an Eisenstein prime (like 11) with no imaginary part and real part of the form .[1] It 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).[15] 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 a regular three-dimensional regular Platonic solid can have, as represented in the regular dodecahedron. In general, a conic curve will require five points in the same way that two points are needed to determine a line.[16]