Perfect numbers

Last month I wrote about happy numbers and sexy numbers so I thought I’d complete a little hat trick and talk all about perfect numbers.

A number is said to be perfect if it equals the sum of its divisors. The second perfect number (28) is divisible by 1,2,4,7,14. 1+2+4+7+14=28

Perfect numbers are rare. At the last count only 48 have been discovered. It is not known whether there are an infinite number of perfect numbers or that if there are any odd ones (Every one found so far is even) and these remain open questions to be solved

Perfect numbers are also entwined with prime numbers as every even perfect number can be represented by the form 2^n − 1(2ⁿ − 1), where 2ⁿ − 1 is a prime number (otherwise known as a Mersenne prime). In order for 2ⁿ − 1 to be a prime number n must be a prime number. This is known as the Euclid–Euler theorem.

Just to get my retaliation in early this time there are no uses whatsoever for perfect numbers (It doesn’t mean one won’t be found, prime numbers were nothing more than a curiosity for centuries) but who says everything we do in life has to be utilitarian? Let’s just marvel at the beauty, interesting nature and perfectness of a small list of numbers.

Happy numbers

A couple of weeks ago I talked about sexy numbers but what makes a number happy? To find out if a number is a happy number you go through the following process.

Pick a number (Lets say 23) square each digit and add them together. so 4+9=13. Now repeat the process.

1+9=10
1+0=1

We've taken a number and through this procedure we have reduced it to 1 so it is now happy. If the number never reaches one then it is an unhappy number. Poor thing :-(

We can also have happy prime numbers such as 313, 331, 367, 379. these numbers are also interesting for another reason to this number geek. There are many happy primes before these numbers but in the Doctor Who episode 42 the numbers 313, 331, 367, 379 are used for unlocking a sealed door but no one can spot the pattern except The Doctor.....see, you never know when this trivia may come in useful!

199th day of the year - A very interesting number

For those who follow me on other social media platforms you’ll have seen my musings on why 197 is an interesting number. Well, as it turns out 199 is also an interesting number.

199 is a twin prime (Twinned with the aforementioned 197) Twin primes are prime numbers that are separated from each other by two….Not as exciting as sexy primes but mathematically more challenging!

It is an 'invertible prime.’ If you turn the number upside down it reads 661 which is also a prime number.

It’s also a ‘circular prime.’ It stays a prime number no matter how you arrange its digits (919 and 991 are also prime numbers)

199 is the sum of 3 consecutive primes 61 + 67 + 71 (Incidentally 661 is also the sum of three prime numbers 211 + 223 + 227)….And also the sum of 5 consecutive primes 31 + 37 + 41 + 43 + 47

Aaaaand if you add its digits up the sum is 19 which is also a prime number.

Exciting eh! :-)

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Oh go on then since I didn’t print it here, this is what is interesting about 197

As well as being a prime number 197 is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37

197 is also the sum of all digits of all two-digit prime numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Aaaand....197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41