Euclids perfect numbers, a study in math only.

[postman]

A Perfect Number N is defined as any positive integer where the sum of its divisors, except the number itself, equals the number

Example 6 is divisible by 1, 2 and 3. And the sum of 1,2 and 3 is 6.

The story of perfect numbers began over twenty-three hundred years ago in one of the most influential mathematical works ever published: The Elements. Born around 300 B.C., the Greek mathematician Euclid founded the study of perfect numbers as a teacher at Alexandria under the reign of Ptolemy the First.

Euclid laid out the basics of perfect numbers over 2,000 years ago, and he knew that the first four perfect numbers were 6, 28, 496 and 8,128.

The problem is not enough people know enough math to discuss it, without wandering off topic. But we will see.

 

[Donald H]

A Perfect Number N is defined as any positive integer where the sum of its divisors, except the number itself, equals the number

Example 6 is divisible by 1, 2 and 3. And the sum of 1,2 and 3 is 6.

The story of perfect numbers began over twenty-three hundred years ago in one of the most influential mathematical works ever published: The Elements. Born around 300 B.C., the Greek mathematician Euclid founded the study of perfect numbers as a teacher at Alexandria under the reign of Ptolemy the First.

Euclid laid out the basics of perfect numbers over 2,000 years ago, and he knew that the first four perfect numbers were 6, 28, 496 and 8,128.

The problem is not enough people know enough math to discuss it, without wandering off topic. But we will see.

Do Euclid's perfect numbers have any significance, say compared to the Fibonacci sequence?

Fibonacci sequence - Wikipedia

en.wikipedia.org

Do you know enough math to say? I don't but I'm interested enough to ask.

 

[postman]

Do Euclid's perfect numbers have any significance, say compared to the Fibonacci sequence?

Fibonacci sequence - Wikipedia

en.wikipedia.org

Do you know enough math to say? I don't but I'm interested enough to ask.

They do have a use in cryptography.

Prime Numbers in Cryptography

Prime Numbers have fascinated mathematicians for a long time. The earliest documented history dates from 500 BCE to 300 BCE when Greek mathematicians studied prime numbers.

www.linkedin.com

Similar to the ways large primes can be used.

 

[postman]

Do Euclid's perfect numbers have any significance, say compared to the Fibonacci sequence?

Fibonacci sequence - Wikipedia

en.wikipedia.org

Do you know enough math to say? I don't but I'm interested enough to ask.

Fibonacci is where the next number is the sum of the previous two, starting with 1 and 2 makes 3
If graphed out, it forms a fractal like spiral pattern.

 

[Donald H]

They do have a use in cryptography.

Prime Numbers in Cryptography

Prime Numbers have fascinated mathematicians for a long time. The earliest documented history dates from 500 BCE to 300 BCE when Greek mathematicians studied prime numbers.

www.linkedin.com

Similar to the ways large primes can be used.

Sorry, the prime numbers aren't Euclid's perfect numbers.

The problem is not enough people know enough math to discuss it, without wandering off topic. But we will see.

LOL

 

[postman]

Sorry, the prime numbers aren't Euclid's perfect numbers.

Ah,,, but that's where you are partly wrong. Perfect numbers factor into powers of two, times a prime number. Or at least do so for perfect numbers that can be computed by formula.

 

[postman]

The first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.

As you see, they get very big, very fast

 

[Donald H]

Ah,,, but that's where you are partly wrong. Perfect numbers factor into powers of two, times a prime number. Or at least do so for perfect numbers that can be computed by formula.

You thought that those perfect numbers were prime numbers. You're not just partially wrong!

 

[postman]

You thought that those perfect numbers were prime numbers. You're not just partially wrong!

Perfect numbers are never prime, because by definition as the sum of it's factors. A prime number is only divisible by 1 and itself. So the sum of its factors (excluding itself) is always 1.

Even if you included itself as a factor, then the sum would be one greater than the number. Hence primes can never be perfect.

 

[postman]

You thought that those perfect numbers were prime numbers. You're not just partially wrong!

I didn't say perfect numbers were prime, but that they always have a prime number as a factor.

 

[ReinyDays]

They do have a use in cryptography.

No ... maybe cryptology ... if the codes are based on perfect numbers ... but otherwise no, we don't use this math for encryption ... I'm not even going to try these sequence-bait ... but the point is good, perfect numbers aren't real useful ...

I'm old enough to remember when the sixth perfect number was discovered ... some ungodly long string of digits I forget how long, but it was newsworthy as to demonstrate the abilities of these new fangled "Gigascale" computers of the day ... and maybe some day in the distant future, this type of reseach will lead to the ability to crack crypto codes, or some such ...

Like hyperbolic sine functions ... no one cares ...

 

[postman]

No ... maybe cryptology ... if the codes are based on perfect numbers ... but otherwise no, we don't use this math for encryption ... I'm not even going to try these sequence-bait ... but the point is good, perfect numbers aren't real useful ...

I'm old enough to remember when the sixth perfect number was discovered ... some ungodly long string of digits I forget how long, but it was newsworthy as to demonstrate the abilities of these new fangled "Gigascale" computers of the day ... and maybe some day in the distant future, this type of reseach will lead to the ability to crack crypto codes, or some such ...

Like hyperbolic sine functions ... no one cares ...

Things always seem useless until somebody discovers a use for them.

The formula basis for perfect numbers are rooted in Mersenne primes, which as large prime numbers are used for encryption, since the product of two large primes can only be factored by a large prime number, which makes finding the factors computationally intensive, hence difficult to crack.

 

[jwoodie]

Six (hexagon) is a perfect number for Geometry and the Physical Sciences.

 

[The Sage of Main Street]

No ... maybe cryptology ... if the codes are based on perfect numbers ... but otherwise no, we don't use this math for encryption ... I'm not even going to try these sequence-bait ... but the point is good, perfect numbers aren't real useful ...

I'm old enough to remember when the sixth perfect number was discovered ... some ungodly long string of digits I forget how long, but it was newsworthy as to demonstrate the abilities of these new fangled "Gigascale" computers of the day ... and maybe some day in the distant future, this type of reseach will lead to the ability to crack crypto codes, or some such ...

Like hyperbolic sine functions ... no one cares ...

Mind Candy for Useless Escapist Nerds

 

[The Sage of Main Street]

Things always seem useless until somebody discovers a use for them.

The formula basis for perfect numbers are rooted in Mersenne primes, which as large prime numbers are used for encryption, since the product of two large primes can only be factored by a large prime number, which makes finding the factors computationally intensive, hence difficult to crack.

The Ivory Tower's Dungeon

If it took 2300 years to find a use for it, it's useless. Think of all the practical payoffs that could have been discovered if geniuses hadn't been distracted by this game.

 

[ReinyDays]

Things always seem useless until somebody discovers a use for them.

The formula basis for perfect numbers are rooted in Mersenne primes, which as large prime numbers are used for encryption, since the product of two large primes can only be factored by a large prime number, which makes finding the factors computationally intensive, hence difficult to crack.

Right ... encryption schemes that use Mersenne primes are broken using Mersenne primes ... I get that ... but what about schemes that use different algorithms? ...