On a curious property of 3435

On a curious prop ert y of 3435. Daan v an Berk el No vem b er 26, 2024 Abstract F ol klore tells us that there are no uninteresting natural n umbers. But so me natural n um b ers are more i nteresti n g then others. In t his article w e will explain wh y 3435 is one of the more interesti n g natural n umb ers around. W e will sh o w that 3435 is a Munchausen numb er in base 10, and w e will explain what w e mean b y that. W e will fu rther sho w that for ev ery base there are finitely man y Munc hausen num b ers i n that base. F olklore tells us that there are no unin teresting natural n um b ers. The argu- men t hinges on the followi ng observ ation: Every subset of the natur al num- b e rs is either empty, or has a smal lest element . The argumen t usually go es something lik e t his. If there would b e any unin teresting natural n um b ers, the set U of all these unin teresting natural n um b ers w ould ha v e a smallest elemen t, sa y u ∈ U . But u in it self has a v ery remark able prop ert y . u is the smallest uninte resting natural num b er, whic h is v ery in teresting indeed. So U , the s et of all the unin teresting natural n um b ers, can not ha v e a smallest elemen t, therefore U m ust b e empty . In other w ords, all nat ura l num b ers are in teresting. Ha ving established this result, exhib iting a n in teresting pro p ert y of a sp e cific natural n um b er is often left as an excercise for the reader. T ak e for example the integer 34 3 5. A t first it do es not seem that remark able, un til one stum bles up on the follow ing iden tit y . 3435 = 3 3 + 4 4 + 3 3 + 5 5 This coincide nce is ev en more remark able when one disco v ers t hat there is only one other natural num b er whic h shares this prop ert y with 3 4 35, namely 1 = 1 1 . In this article w e will establish the claim made and generalize the result. 1 Munc hausen Num b e r Through o ut the article w e will use the follo wing notation. b ∈ N will de- notate a base and therefore the inequilit y b ≥ 2 will hold throughout the article. F or ev ery natural n umber n ∈ N , the b ase b r e p r esentation of n will b e denoted by [ c m − 1 , c m − 2 , . . . , c 0 ] b , so 0 ≤ c i < b fo r all i ∈ { 0 , 1 , . . . , m − 1 } and n = P m − 1 i =0 c i b i . F urtheremore , w e define a function θ b : N → N : n 7→ P m − 1 i =0 c c i i , where n = [ c m − 1 , c m − 2 , . . . , c 0 ] b . W e will further ado pt the conv en- tion that 0 0 = 1, in accordance with 1 0 = 1, 2 0 = 1 etcetera. Definition An integer n ∈ N is called a Munchausen numb er in b ase b if and only if n = θ b ( n ). ◦ So by the equalit y in the in tro duction w e kno w that 3435 is a Munc hausen n um b er in base 10 . Remark A related concept to Munc hausen n um b er is that of Narcissistic n um b er. (See for example [1], [2] and [3].) The reason fo r pic king the name Munc hausen n um b er stems f ro m the visual of raising oneself, a feat demonstrated by the famous Baron v on Munc hausen ([4]). Andrew Baxter remark ed that the Bar o n is a narcissistic man indeed, so I think the name is aptly chose n. ⊳ The f ollo wing tw o lemmas will b e used to pro of the main result of this ar- ticle: for eve ry base b ∈ N there a re only finitely man y Munc hausen n um b ers in base b . Lemma 1 F or al l n ∈ N : θ b ( n ) ≤ (log b ( n ) + 1)( b − 1) b − 1 . ⋄ Pro of Notice that the function x 7→ x x is strictly increasing if x ≥ 1 e . This can b e seen from the deriv ativ e of x x whic h is x x (log( x ) + 1). This last express ion is clearly p ositiv e fo r x > 1 e . T o g e ther with the definition of 0 0 = 1, w e see that x x is increasing for all the nonnegativ e in tegers. F or a ll n ∈ N with n = [ c m − 1 , c m − 2 , . . . , c 0 ] b w e hav e the ineqalities 0 ≤ c i ≤ b − 1 for all i within 0 ≤ i < m . So θ b ( n ) = P m − 1 i =0 c c i i ≤ P m − 1 i =0 ( b − 1) b − 1 = m × ( b − 1) b − 1 . No w, the num b er of digits in the base b represan tation of n equals ⌊ log b ( n ) + 1 ⌋ . In other w ords m := ⌊ log b ( n ) + 1 ⌋ ≤ log b ( n ) + 1. So θ b ( n ) ≤ (log b ( n ) + 1)( b − 1) b − 1  Lemma 2 I f n ∈ N and n > 2 b b then n log b ( n )+1 > ( b − 1) b − 1 . ⋄ 2 Pro of Let n ∈ N suc h that n > 2 b b . Notice that x 7→ x log b ( x ) is strictly increasing if x > e . T o see this notice that the deriv a t ive of x log b x is log( b ) log( x ) − 1 log 2 ( x ) whic h is p ositiv e for x > e . F urthermore log b (2) + 1 ≤ 2 ≤ b = b log b ( b ). No w, b ecause n > 2 b b > e , from the following c hain of ineqalities: n log b ( n ) + 1 > 2 b b b log b ( b ) + log b (2) + 1 ≥ 2 b b 2 b log b ( b ) = b b − 1 > ( b − 1) b − 1 w e can deduce that n log b ( n )+1 > ( b − 1) b − 1  With b oth lemma’s in place w e can presen t without f urthe r ado the main result of this article. Prop osition 3 F or every b a s e b ∈ N with b ≥ 2 : ther e ar e on l y finitely many Munchausen numb ers in b ase b . ⋄ Pro of By the precedin g lemma’s w e ha v e, for all n ∈ N with n > 2 b b : n > (log b ( n ) + 1)( b − 1) b − 1 ≥ θ b ( n ). So, in order for n to equal θ b ( n ), n mus t be less then or equal to 2 b b . This pro ve s that there are only finitely many Munc hausen nu mbers in base b .  Exhaustiv e Searc h The prop osition in the preceding section tells use that fo r ev ery base b ∈ N , Munc hausen n um b ers in that base only o ccur within the in terv al [1 , 2 b b ]. This mak es it p ossible to exhaustiv ely searc h fo r Munc hausen n um b ers in eac h base. Figure 1 lists all the Munc hausen n um b ers in the bases 2 through 10. So for example in base 4, 2 9 and 55 are the only non-trivial Munc hausen n um b ers. F urthermore, the base 4 represen t a tion of 29 and 55 hav e a striking resem blance. F or 29 = [1 , 3 , 1 ] 4 = 1 1 + 3 3 + 1 1 and 55 = [3 , 1 , 3] 4 = 3 3 + 1 1 + 3 3 . The sequence of Munc hausen num b ers is listed as seq uence A166623 at the OEIS. (See [5]. F or the r elated sequence o f Narcissistic nu mbers see [6]) The co de in listing 1 is used to pro duce the n umbers in figure 1. There are tw o utilit y f unctions. These a re m unc hausen and next. m unc hausen calculates θ b ( n ) give n a base b represen tation of n . next returns the base b represen ta tion of n + 1 giv en a base b represen tation of n . I w ould like to conclude this article with a question my wife ask ed me while I w as writing this: “But what ab out 20082009?” 3 Figure 1: Munc hausen nu mbers in base 2 through 10. Base Munc hausen Num b ers Represen tation 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 co de finding Munc hausen n um b ers n e x t : = f u n c t i o n ( c o e f f i c i e n t s , b ) l o c a l i ; c o e f f i c i e n t s [ 1 ] : = c o e f f i c i e n t s [ 1 ] + 1 ; i : = 1 ; w h i l e c o e f f i c i e n t s [ i ] = b do c o e f f i c i e n t s [ i ] := 0 ; i := i + 1 ; i f ( i < = L e n g t h ( c o e f f i c i e n t s ) ) th en c o e f f i c i e n t s [ i ] : = c o e f f i c i e n t s [ i ] + 1 ; e l s e Add ( c o e f f i c i e n t s , 1 ) ; f i ; od ; r e t u r n c o e f f i c i e n t s ; end ; m u n c h a u s en : = f u n c t i o n ( c o e f f i c i e n t s ) l o c a l sum , c o e f f i c i e n t ; sum : = 0 ; f o r c o e f f i c i e n t i n c o e f f i c i e n t s do sum : = sum + c o e f f i c i e n t ˆ c o e f f i c i e n t ; od ; r e t u r n s u m ; end ; f o r b i n [ 2 . . 1 0 ] do ma x : = 2 ∗ b ˆ b ; n : = 1 ; c o e f f i c i e n t s : = [ 1 ] ; w h i l e n < = max do sum : = m u n c h a u s e n ( c o e f f i c i e n t s ) ; i f ( n = sum ) t h en P r i n t ( n , ” \ n ” ) ; f i ; n : = n + 1 ; c o e f f i c i e n t s : = n e x t ( c o e f f i c i e n t s , b ) ; od ; od ; 4 References [1] Clifford A. Pic k o ver. Wonders o f Numb ers . Oxford Unive rsity Press, 2001. [2] Wikip edia. Narcissistic Num b er. http://en .wikipedia.org / wiki/N a rci s si st i c_ n umber . [3] W olfram Math W orld. Narcissistic Num b er. http://m athworld.wolfra m.com/NarcissisticNumber.html . [4] Wikip edia. Baron Munc hhausen. http://e n.wikipedia.or g/wiki/Baron_Munchhausen . [5] The On-Line Encyclopedia of In teger Sequences. A166623. http://w ww.research.att .com/ ~ njas/seq uences/A166623 . [6] The On-Line Encyclopedia of In teger Sequences. A005188. http://w ww.research.att .com/ ~ njas/seq uences/A005188 . 5