Thursday, June 23, 2022

5555 - 5

The evolution of 5.

How many 5's are there?

You took math in high school, right? Read through this and see how many terms are familiar to you.

Five is the third smallest prime number.[1] Since it can be written as 22 + 1, five is classified as a Fermat prime;[1] therefore, a regular polygon with 5 sides (a regular pentagon) is constructible with compass and an unmarked straightedge. Five is the third Sophie Germain prime,[1] the first safe prime, the third Catalan number,[2] and the third Mersenne prime exponent.[3] Five is the first Wilson prime and the third factorial prime, also an alternating factorial.[4] Five is the first good prime.[5] It is an Eisenstein prime with no imaginary part and real part of the form 3p − 1.[1] It is also the only number that is part of more than one pair of twin primes. Five is also a super-prime, and a congruent number.[6]

Five is conjectured to be the only odd untouchable number,[7] and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.

Five is also the only prime that is the sum of two consecutive primes, namely 2 and 3, with these indeed being the only possible set of two consecutive primes.

The number 5 is the fifth Fibonacci number, being 2 plus 3.[1] It is the only Fibonacci number that is equal to its position. Five 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), ... (OEISA030452 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.[8]

5 is the length of the hypotenuse of the smallest integer-sided right triangle.

In bases 10 and 20, 5 is a 1-automorphic number.

Five is the second Sierpinski number of the first kind, and can be written as S2 = (22) + 1.[9]

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.

There are five known unitary perfect numbers. A sixth unitary number, if discovered, would have at least nine odd prime factors.[10]

There are five Platonic solids.[11][1] There are also five regular uniform polyhedron compounds,[12] five cell-transitive space-filling parallelohedra,[13] and five abstract polyhedra.[14]

There are five regular honeycombs in hyperbolic 5-space. [15]

polygon with five sides is a pentagonFigurate 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, unlike expansions with all other prime denominators, 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.

To see even more about 5...


Elephant's Child said...

My brain hurts. Five times as much as when I started to read your post.

Mike said...

Sue - Is it a Platonic solids hurt or an Abel–Ruffini theorem hurt?

River said...

I did not know five was/is a prime number. I didn't understand any of the rest of that, having left school early to get a paying job instead.

Mike said...

River - Hopefully not a job as a cashier.

Bilbo said...

But is it an Amazon Prime, deliverable on the next day?

Debra She Who Seeks said...

Math gives me hives.

Kathy G said...

Bilbo's comment made me laugh out loud!

John A Hill said...

My favorite primes are Amazon Prime and prime rib (medium rare with a little horseradish, please).

Mike said...

Bill - Only on days ending in 5.

Deb - Alright! We need pictures!

Kathy - Don't encourage him. Once he gets on a roll you can't stop him.

John - AP yes for me. But I'm a filet guy. And NO horse radish!

Mr. Shife said...

You lost me at math in high school.

Mike said...

Matt - I just found you in spam jail. High school math was not so bad for me. In college, calculus II did me in.

River said...

The cashier job came later, I began in a dairy factory, bottling milk and wrapping 40 pound cheeses for export, then moved to shoe factories, before finally being a supermarket checkout operator for twelve years.

Cloudia said...

Wow 🥺

Mike said...

River - I've got a longer trail than that.

Cloudia - Had you ever heard of any of this crazy stuff? Me neither.

jenny_o said...

I know a lot of those words, like "the", "with", "is", and best of all, "five". Hah! Take that :)

Mike said...

Jenny - Well you know Five is conjectured to be the only odd untouchable number? Are you an odd untouchable number?