Hey Steemians! My steemit rank has hit 42 so today we will be looking at 42 the most famous number in geek culture. Nerds will know 42 as the Answer to the Ultimate Question of Life, The Universe, and Everything from the magnificent book the Hitchhiker's guide to the galaxy by Douglas Adams. The number 42 is actually not perfect but only a primary pseudoperfect number. So in this post we will delve into the world of perfect and primary pseudoperfect numbers and explore where 42 fits in.

Perfect numbers

We start our journey with perfect numbers. These numbers go all the way back to antiquity. So what are perfect numbers? A perfect number is a whole number which is equal to the sum all numbers by which you can divide it (and get a whole number back) excluding the number itself. The smallest perfect number is 6. Let us check that 6 is a perfect number: all its divors excluding 6 are given by 1,2,3 which when added up together give 1+2+3=6. The next perfect number is 28 since all its divisors except 28 are given by 1,2,4,7,14 which when added up give 1+2+4+7+14=28. Well with a lot of calculations you can find that 496 is the next number and after that we have 8128.

So our perfect number list is 6,28,496,8828. The remarkable property which already becomes apparent is that they all end in 6 or 8. You can show (quite easily) that all even perfect numbers end in a 6 or 8[1]. However, it is unknown if all perfect numbers are even. Furthermore, only 50 perfect numbers are known[2]. So maybe there is an odd perfect number out there. Maybe there are only 50 perfect numbers? Mathematician are unable to prove that there are infintely many perfect numbers so perfect numbers give rise to many unanswerd questions.

Why study perfect numbers?

There is a direct connection between perfect numbers and a special type of prime numbers called Mersenne primes. We call p a Mersenne prime if it is a prime and if there exists a whole number k such that

It is easy to see that 3 is a Mersenne prime. This is then followed by 3, 7, 31,127,... Observe that (1) does not give a prime number for any k. For example take k=4 then we get that 2^4-1=15. Out of the blue let us take the Mersenne primes 3, 7, 31,127 and solve some simple equations:

3(3+1)/2=6
7(7+1)/2=28
31(31+1)/2=496
127(127+1)/2=8828

OMG all of these numbers are perfect number! With a little effort you can prove that this is true for all (even) perfect numbers[5]. This creates a link between Mersenne primes and perfect numbers. So we can use perfect numbers to generate primes. The Equation (1) admits very large primes. Large primes are usefull for cryptography so that is why there is a lot of interest for perfect numbers.

Primary pseudoperfect numbers

So what are primary pseudoperfect numbers? Avoiding complicated notation let me first define primary pseudoperfect number in words and then give some examples.

A number q is called a primary pseudoperfect number when you take all the primes divisors of q and q itself, divide one by each number and then add them all together get one. I know this is a mouthfull so let's take a look at a few examples. 2 is the smallest primary pseudoperfect number. We know that 2 divides 2 and that it is a prime. Note that mathematicians do not consider 1 a prime number. We have that

So 2 is a primary pseudoperfect number. The number 6 is also a primary pseudoperfect number. Its prime divisors are given by 2,3. We have that



Making it primary pseudo perfect. The next primary pseudoperfect number is 42 since its prime divisors are given by 2,3,7 and we have that



Making 42 an extra cool number. With a lot of effort you can find that 1806 is also a pseudoperfect number. So our list is 2,6,42,1806. Primary pseudoperfect numbers, like perfect numbers, are shrouded in mystery. Although from our list it seems like they should be even numbers this is unknown [3]. So far all the computed primary pseudoperfect numbers turned out to be even[4]. It is also unknown if there are infinitely many primary pseudoperfect numbers[3].

Is there a connection between primary pseudoperfect numbers and perfect numbers?

You can see some similarity between how primary pseudoperfect numbers and perfect numbers are defined. More specifically, primary pseudoperfect numbers inherited their name from perfect numbers. Primary pseudoperfect numbers are young for mathematics standard since they were only defined in 2000[3]. So they were only created after Hitchhiker's guide to the Galaxy was written. The unsolved problems connected to both numbers are of the same type. However, in the research field there does not seem to be a connection (yet). But maybe in the future a connection will be found? Maybe 42 will be at the center of this discovery ;)

Sources

[1] http://mathworld.wolfram.com/PerfectNumber.html
[2] https://en.wikipedia.org/wiki/List_of_perfect_numbers
[3] https://en.wikipedia.org/wiki/Primary_pseudoperfect_number
[4] https://oeis.org/A054377
[5] https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler_theorem

Top picture by Martinultima - Image:Answer to Life.png, CC BY-SA 3.0

---

Further reading/watching

If you are interested more about perfect number you can check the wiki page and if you are interested in the proof that gives the connection between Mersenne primes you can check this video over here

---

Thank you!

Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below :o)

---

Owl tax

So long and thanks for all the fish!

Source