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Through out the article we will use the following notation. b ∈ N will de- notate a base and therefore the inequility b ≥ 2 will hold throughout the article. For every n ∈ N, the base b representation of n will be denoted by [cm−1,cm−2,...,c0]b,so 0 ≤ ci < b for all i ∈{0, 1,...,m − 1} m−1 i N N and n = Pi=0 cib . Furtheremore, we define a function θb : → : n → m−1 ci Pi=0 ci , where n = [cm−1,cm−2,...,c0]b. We will further adopt the conven- tion that 00 = 1, in accordance with 10 =1, 20 = 1 etcetera.

Definition An integer n ∈ N is called a Munchausen number in base b if and only if n = θb(n). ◦

So by the equality in the introduction we know that 3435 is a Munchausen number in base 10.

Remark A related concept to Munchausen number is that of Narcissistic number. (See for example [1], [2] and [3].) The reason for picking the name Munchausen number stems from the visual of raising oneself, a feat demonstrated by the famous Baron von Munchausen ([4]). Andrew Baxter remarked that the Baron is a narcissistic man indeed, so I think the name is aptly chosen. ⊳

The following two lemmas will be used to proof the main result of this ar- ticle: for every base b ∈ N there are only finitely many Munchausen numbers in base b.

b−1 Lemma 1 For all n ∈ N: θb(n) ≤ (logb(n)+1)(b − 1) . ⋄

x 1 Proof Notice that the function x → x is strictly increasing if x ≥ e . This can be seen from the derivative of xx which is xx(log(x) + 1). This last 1 expression is clearly positive for x> e . Together with the definition of 00 = 1, we see that xx is increasing for all the nonnegative .

For all n ∈ N with n = [cm−1,cm−2,...,c0]b we have the ineqalities 0 ≤ ci ≤ b − 1 for all i within 0 ≤ i < m. m−1 ci m−1 b−1 b−1 So θb(n)= Pi=0 ci ≤ Pi=0 (b − 1) = m × (b − 1) . Now, the number of digits in the base b represantation of n equals ⌊logb(n)+1⌋. In other words m := ⌊logb(n)+1⌋ ≤ logb(n)+1. b−1 So θb(n) ≤ (logb(n)+1)(b − 1) 

Lemma 2 If n ∈ N and n> 2bb then n > (b − 1)b−1. ⋄ logb(n)+1

2 Proof Let n ∈ N such that n > 2bb. Notice that x → x is strictly logb(x) increasing if x > e. To see this notice that the derivative of x is logb x log(x)−1 b 2 x > e log( ) log (x) which is positive for . Furthermore logb(2) + 1 ≤ 2 ≤ b = b logb(b). Now, because n> 2bb > e, from the following chain of ineqalities: n 2bb 2bb > ≥ = bb−1 > (b − 1)b−1 logb(n)+1 b logb(b)+logb(2)+1 2b logb(b) we can deduce that n > (b − 1)b−1  logb(n)+1 With both lemma’s in place we can present without further ado the main result of this article.

Proposition 3 For every base b ∈ N with b ≥ 2: there are only finitely many Munchausen numbers in base b. ⋄

Proof By the preceding lemma’s we have, for all n ∈ N with n > 2bb: b−1 n> (logb(n)+1)(b − 1) ≥ θb(n). b So, in order for n to equal θb(n), n must be less then or equal to 2b . This proves that there are only finitely many Munchausen numbers in base b. 

Exhaustive Search

The proposition in the preceding section tells use that for every base b ∈ N, Munchausen numbers in that base only occur within the interval [1, 2bb]. This makes it possible to exhaustively search for Munchausen numbers in each base. Figure 1 lists all the Munchausen numbers in the bases 2 through 10. So for example in base 4, 29 and 55 are the only non-trivial Munchausen numbers. Furthermore, the base 4 representation of 29 and 55 have a striking 1 3 1 3 1 3 resemblance. For 29 = [1, 3, 1]4 =1 +3 +1 and55 = [3, 1, 3]4 =3 +1 +3 . The sequence of Munchausen numbers is listed as sequence A166623 at the OEIS. (See [5]. For the related sequence of Narcissistic numbers see [6]) The code in listing 1 is used to produce the numbers in figure 1. There are two utility functions. These are munchausen and next. munchausen calculates θb(n) given a base b representation of n. next returns the base b representation of n + 1 given a base b representation of n. I would like to conclude this article with a question my wife asked me while I was writing this: “But what about 20082009?”

3 Figure 1: Munchausen numbers in base 2 through 10. Base Munchausen Numbers Representation 2 1, 2 [1]2, [1, 0]2 3 1, 5, 8 [1]3, [1, 2]3, [2, 2]3 4 1, 29, 55 [1]4, [1, 3, 1]4, [3, 1, 3]4 5 1 [1]5 6 1, 3164, 3416 [1]6, [2, 2, 3, 5, 2]6 , [2, 3, 4, 5, 2]6 7 1, 3665 [1]7, [1, 3, 4, 5, 4]7 8 1 [1]8 9 1, 28, 96446, 923362 [1]9, [3, 1]9, [1, 5, 6, 2, 6, 2]9 , [1, 6, 5, 6, 5, 4, 7]9 10 1, 3435 [1]10, [3, 4, 3, 5]10

Listing 1: GAP code finding Munchausen numbers next := function (coefficients , b) local i ; coefficients[1] := coefficients[1] + 1; i := 1; while coefficients[i] = b do coefficients[i] := 0; i := i + 1; if (i <= Length(coefficients )) then coefficients[i] := coefficients[i] + 1; else Add(coefficients , 1); fi ; od ; return coefficients ; end ; munchausen := function (coefficients) local sum, coefficient; sum := 0; for coefficient in coefficients do sum := sum + coefficientˆcoefficient; od ; return sum; end ; for b in [2..10] do max := 2∗bˆb; n := 1; coefficients := [1]; while n <= max do sum := munchausen(coefficients );

if (n = sum) then Print(n, ”\n”); fi ;

n := n+ 1; coefficients := next(coefficients , b); od ; od ;

4 References

[1] Clifford A. Pickover. Wonders of Numbers. Oxford University Press, 2001.

[2] Wikipedia. Narcissistic Number. http://en.wikipedia.org/wiki/Narcissistic_number.

[3] Wolfram Math World. Narcissistic Number. http://mathworld.wolfram.com/NarcissisticNumber.html.

[4] Wikipedia. Baron Munchhausen. http://en.wikipedia.org/wiki/Baron_Munchhausen.

[5] The On-Line Encyclopedia of Integer Sequences. A166623. http://www.research.att.com/~njas/sequences/A166623. [6] The On-Line Encyclopedia of Integer Sequences. A005188. http://www.research.att.com/~njas/sequences/A005188.

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