The equation for PHI can be written exclusively using the number 5

Note also that the Pentagon has close relationships with Φ. Learn how to construct a Pentagon in the article Pentagon elsewhere in this category.

The relationship between PHI and the Pentagon is touched upon in the article PHI in the same category.

The Greek word for five is PENTE, which itself is derived from PAN, who represents nature.

The name quincunx is the name for the '5' pattern found on a die. It's also the name for a particular type of 5-domed cathedral, like St Mark's Cathedral in Venice and certain Khmer temples in South East Asia.

Following excerpts are adapted from the book Wonders of Numbers by Clifford A. Pickover (ISBN: 0-19-515799-0).

Not only is 5 the hypotenuse of the smallest Pytagorean triangle, but it is also the smallest automorphic number. Let me explain. A Pytagorian triangle is a right-angled triangle with integral sides. For example, the smallest Pytagorean triangle has side lengths 3, 4, and 5. An automorphic number n, when multiplied by itself, leads to a product whose rightmost digits are n. Not counting the trivial case of number 1, the numbers 5 and 6 are the smallest automorphic numbers because 5 x 5 = 25 and 6 x 6 = 36. Examining a larger number, the square of 25 is 625. Note that 25 appears as the final 2 digits of 625.

Only 2 numbers divide 10: 5 and 2.
Five is probably the only odd untouchable number. The legendary and bizarre mathematician Paul Erdos called a number untouchable if it is never the sum of ALL the proper divisors of any other number. A divisor of a number N, is a number d which divides N (also called a factor). A proper divisor is simply a divisor of the number N, excluding N itself.
For example, 4 is not untouchable as it can be made up of the sum of the proper divisors of 9, i.e. 1 and 3.
The sequence of untouchable numbers starts 2, 5, 52, 88, 96, 120.

There are 5 Platonic solids: The tetrahedron, cube, octahedron, dodecahedron, and icosahedron. All the faces of a Platonic solid must be congruent regular polygons.

The real meaning of 5 in the Golden ratio is that of ‘half 10’, i.e. the proportion of a half and the square root of 5, i.e the number that, multiplied with itself, gives 5.

The interval of a fifth is referred to in music as the dominant. Moving in progressions of fifths has a driving quality about it. The circle of fifths underlies our musical language, while the pentatonic scale is the most fundamental scale prominent in all cultures. I've gone into depth about this and musical scales in general in the music - music science category.