*Thought it would be fun to write a post about properties of (whole positive) numbers a.k.a. natural numbers. Hope you won't 8 it .*

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Number theory is a branch of mathematics which concerns the study of (surprise surpirse) numbers. More specifically, it concerns whole number.  A great part of number theory involves selecting numbers through an interesting/beautiful/useful property and then showing that these selected numbers satisfy even more  interesting/beautiful/useful properties.

## Let's have a look at some cool numbers:
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**Narcissistic numbers or Munchausen numbers** (named after the German baron with over-the-top adventures) are whole numbers which are equal to the sum of all its digits raised by the corresponding digit. So for example, since 1=1<sup>1</sup> it is a Narcissistic number and 3435=3<sup>3</sup>+4<sup>4</sup>+3<sup>3</sup>+5<sup>5</sup> is also a Narcissistic number. There are only finitely many Narcissistic numbers in our base 10 counting system (=counting with 10 fingers). More generally, it turns out that in any base there are finitely many Narcissistic numbers[1]. I think that that is a pretty neat property.

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**Friedman numbers** are whole numbers which can be expressed as some combination of its own digits with any of the operations +, −, ×, ÷ (note that you are only allowed to use each digit once but you can use an operation as many times as you want). So 25=5<sup>2</sup> is a Friedman number. With some basic math you can show that it is  the smallest Friedman number. You are also allowed to take two digits and turn them into a number, so 126=21 x 6 is also Friedman number. 



There are all kinds of cool subsets of Friedman numbers. An exceptionally nice one are **nice Friedman numbers** which are numbers which preserve the order of the digits, so 343=(3+4)<sup>3</sup> is a nice Friedman number because on the left hand side there first is a 3 then a 4 and finally again a 3. There are infinitely many Friedman numbers [2]. The remarkable thing is that if you consider a random number of n digits then the probability that this is a Friedman number converges to one if you let n go to infinity [2]. So there are many of these fellows if you consider numbers with sufficiently many digits. Unfortunately, this result remains unproven for nice Friedman numbers in base 10. But you can prove in certain lower bases which is nice [3].


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**Perfect numbers** are one my favorite numbers. So a perfect number is a whole number which is equal to the sum of all its divisors while excluding the number itself. For example 6 is a perfect number since its divisor other than 6 are given by 1,2,3 and 1+2+3=6. It is a simple exercise to show that 28 is also perfect number. There are many open problems related to perfect numbers. As computer power has improved we have been able to always construct a new largest perfect number [4]. That raises the question are there infinitely many? This is an open problem. If you take a look at a list of perfect numbers you notice that they are all even [4]. Can we prove this? No, this is also an open problem. However you can prove that perfect numbers  are connected to  a special type of prime number, called Mersenne primes [5] (check out this [Numberphile video](https://www.youtube.com/watch?v=T0xKHwQH-4I) for an easy to understand proof). Which is awesome.


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## Are all numbers interesting?
I could go on all day talking/writing about numbers but this might lead you to asking the question: do I think that all (whole) numbers are interesting? And the answer is yes. Let me prove it to you. Suppose there exists a (non-empty) collection of numbers which are uninteresting. Then there must  exist least interesting numbers which is an interesting property. So there you have it: all numbers are interesting :D 

This is paradoxical statement is also known as the interesting number paradox [6]. (I actually have no idea who came up with it so if you know where it originally came from let me know in the comments below). Since I am proven to be interested in all numbers tell me your favorite number  in the comments below :D 


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### References
[1] D. van Berkel, On a curious property of 3435. https://arxiv.org/pdf/0911.3038v2.pdf 
[2] M. Brand, Friedman numbers have density 1, Discrete Applied Mathematics, 161(16–17), Nov. 2013, pp. 2389-2395.
[3] M. Brand, On the Density of Nice Friedmans, Oct 2013, https://arxiv.org/abs/1310.2390
[4]  M. Pedersen, A list of known perfect numbers https://web.archive.org/web/20090503154707/http://amicable.homepage.dk/perfect.htm
[5] L. Euler (1849), De numeris amicibilibus, Commentationes arithmeticae (in Latin), 2, pp. 627–636. Originally read to the Berlin Academy on February 23, 1747, and published posthumously. See in particular section 8, p. 88.
[6] B. Brunch, Mathematical Fallacies and Paradoxes,  New York 1982.

Remark concerning [1]: The number of Narcissistic numbers actually depends on if you assume that 0<sup>0</sup>=1. If you assume this then for base 10 there exist 4 Narcissistic numbers 0,1, 3435, and 438579088. This paper does not  assume that 0<sup>0</sup>=1 so it says there are only 2 Narcissistic numbers for base 10. 

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**All images made with [inkscape](https://inkscape.org/en/). It is free!**

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<h2> Merchandise :D </h2><br />
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