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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Prime Numbers - Advanced

Ganesh and I have just worked together on creating a page about Primes, Perfect, Abundant and Deficient Numbers, Amicable Numbers, Mersenne's Primes and even Goldbach's Conjecture. Here it is: Prime Numbers - Advanced

This is the first page in the "Advanced" series - you will notice new graphics. "Advanced" may not be the best word, but I can't think of a better one, suggestions welcome.

Enjoy! And feel free to comment or correct, as usual.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

justlookingforthemoment

Moderator

Registered: 2005-05-26

Posts: 2,161

Re: Prime Numbers - Advanced

Excellent!

Advanced is blue!

Maybe you could go on from amicable numbers to have topics on sociable groups and Aliquot sequences?

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Prime Numbers - Advanced

That is such a good idea ... what do you think I should write?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

justlookingforthemoment

Moderator

Registered: 2005-05-26

Posts: 2,161

Re: Prime Numbers - Advanced

Well, I just thought those topics kind of linked...

So, you wanted some ideas :

Amicable numbers - where the sum of one number’s factors is equal to another number, and vice versa.

Sociable groups - this is the same as amicable numbers but with 3, 4, 5 or more! The 'order' of a sociable group is how many number are in the sequence. So, perfect numbers are of order-1, sociable groups of order-2 are amicable numbers. But, no groups of order-3 have been found yet. There are, however, many groups of 4, 5 etc.

Smallest group of order-4
1264460, 1547860, 1727636, 1305184

Of order-5:
12496, 14288, 15472, 14536, 14264

Aliquot sequences - every number has its own Aliquot sequence, and lots of Aliquot sequences are the same - they just start in different places. To find a number's Aliquot sequence first add its proper factors (all positive factors excluding the number itself):

20 has proper factors 1, 2, 4, 5, 10, which add to make 22
22 has proper factors 1, 2, 11 which add to make 14
14 has proper factors 1, 2, 7 which add to make 10
10 has proper factors 1, 2, 5 which add to make 7
7 has proper factors 1 which add to make 1
1 has no proper factors, which add to make 0
0 has no proper factors, which add to make 0 ...

An Aliquot sequence terminates when it reaches a number already found (forming a sociable group) or when 1 is reached, because the next number will automatically be zero.

A perfect number's aliquot sequence consists only of itself.
A prime number's aliquot sequence consists of itself and 1.
An amicable number's aliquot sequence consists of itself and its amicable partner...

There are also Aliquot sequences that go forever (at least, nobody knows when they will stop)!

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,413

Re: Prime Numbers - Advanced

Thanks justlookingforthemoment,
I am sure these are not taught at school!
Wonder where you got them from!

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

justlookingforthemoment

Moderator

Registered: 2005-05-26

Posts: 2,161

Re: Prime Numbers - Advanced

I did a project on topics like these a while ago...