Number Gossip

Num b er Gossip T an ya Kho v anov a Dep artment of Mathematics, MIT Ma y 30, 2018 Abstract This article cov ers my talk at the Gathering for Gardner 2008, with some additions. 1 In tro duction My p et pro ject Number Gossip has its own w ebsite: http://www. numbergossip.com/ , where you can plug in your fa vorite integ er up to 9, 999 and learn its prop erties. A b ehind-the-scenes program chec ks your num b er for 49 regular prop erties and also c h ec ks a database for u nique p roperties I collected. 2 Eigh t The fa vorite num b er of this year’s Gathering is comp osite, deficient, even, odious, palindromic, p o werful, practical and Ulam. It is also very cool as it has the rare prop erties of b eing a cake and a narcissistic num b er, as well as a cub e and a Fib onacci num b er. And it also is a p ow er of tw o. In addition, eight has the follo wing unique prop erties: • 8 is the only comp osite cub e in the Fibonacci sequence • 8 is the dimension of th e o ctonions and is the highest p ossible dimension of a normed division algebra • 8 is the smallest number (except 1) whic h is equal to the sum of the digits of its cub e 3 Prop erties There are 49 regular prop erties that I chec k for: abundant evil odious Smith amicable factorial palindrome sociable apo calyptic p o wer Fibon acci palindromic prime square aspiring Google p en tagonal square-free automorphic happy p erfe ct tetrahedral cake hungry p o wer of 2 triangular Carmic hael lazy caterer p o werf u l t win Catalan lucky practical Ulam composite Mersenne prime undulating compositorial Mersenne prime p rimo rial untouc hable cub e narcissistic pronic v ampire deficient odd repunit w eird even 1 I selected regular prop erties for their importance as w ell as their funny n ames , so your fav orite num b er could b e lucky and happy at th e same t ime, as is the case for 7. Here are some definitions: The number n is called an a pocal yptic pow er if 2 n conta ins 666. The smallest ap ocalyptic p ow er is 157: 2 157 = 182687704666 362864775460604089535377456991567872. The bigger your num b er the more lik ely it is to b e an ap ocalypt ic p ow er. As your numbers gro w the ap ocalypse is b ecomes more and more probable. The num b er n is e vil if it has an even number of ones in its binary expansion. Can you gu ess what o dious num b ers are (see John Conw a y at al. [1])? The n th lazy c aterer n umber is th e maximum n u mb er of pieces a circular pizza can b e cut into with n straigh t- li ne cu ts. F or some reason mathematicians think that pizza is a 2-dimensional ob ject. That means that horizon tal cuts are n ot allo wed. A 3D p izz a is called a cak e, so you can deduce whic h are cake numbers. F or cak e num b ers, h ori zonta l cuts are allow ed, but beware: someone ma y b e deprived of icing. The k th h ungry n umber is the small est n u m b er n suc h that 2 n conta ins the first k digits of th e decimal expansion of π . Here again, math ematicians think that π is edible. On m y website, I div id ed regular prop erties into common p roperties and rare prop ert ies. This separation can increase yo ur pride if your fav orite num b er turns out to have a rare prop ert y . There are few er than 100 num b ers b elo w 10,000 that h a ve one of these rare properties. At 140 below 10,000, lazy caterer and triangular num b ers are not quite rare enough to b e in this rare category , whereas pronic and sq uare n umb ers, with 99 b elo w 10,000, just mak e it onto the rare side. Rare prop erties are: amicable, aspiring, automorphic, cake, Carmichael, Catalan, comp ositori al, cub e, facto- rial, Fib onacci, Go ogl e, hungry , Mersenne, Mersenne p rime, narcissistic, p al indromic prime, p en tagonal, p erfect, p o wer of 2, primorial, pronic, repunit, square, tetrahedral, v ampire, wei rd. 4 Unique Prop erties Currently I hav e uploaded 839 uniq ue prop erties. H ere are some of my fav orites: 11 is the largest num b er whic h is not expressible as the sum of t wo composite num b ers. 19 is the largest prime which is a palindrome in Roman numerals. 27 is the only num b er whic h is thrice th e sum of its d ig its. 27 is the largest num b er that is the sum of the digits of its cub e. 38 has a represen t atio n in Roman numeral s — XXXV I I I — that is lexicographically th e last p ossible Roman numeral. 40 when written in English “forty” is the only num b er whose constituent letters app ear in alphab etical order. 99 is the larges t number that is equal t o the sum of its digits plus the pro duct of its digits: 99 = 9 + 9 + 9 * 9. 119 cents is the largest amount of money one can hav e in coins without b ei ng able to make c hange for a dollar. 144 is the only comp osite square in the Fib onacci sequence. 888 is the only num b er whose cub e, 700227072, consists of 3 digits each occurring 3 times. 1089 is the smallest num b er whose reverse is a non-trivial integer multiple of itself: 9801 = 9*1089 . 1210 is the smallest autobiographical num b er: n = x 0 x 1 x 2 · x 9 such that x i is the num b er of digits equal t o i in n : 1210 is built of one 0, tw o 1s, and one 2. Martin Gardner in his b ook “Mathematical Circus” [3] gives a puzzle to calculate t h e only 10-digit autobiographical number. See also my pap er [4]. 1331 is the smallest non-t riv ia l cub e containing only odd d igi t s. Here are some prop erties that are easy to prov e that you can do as an exercise: 5 is the only prime which is the difference of tw o squares of primes. 2 6 is the only mean b et ween a pair of twi n primes which is triangular. 1728 is the only comp osito rial cub e (submitted by Sergei Bernstein). I started this website for c hildren, so my fi rst prop erties w ere very easy and w ell-k no wn like: 5 is the num b er of Platonic solids. 9 is the smallest o d d composite number. At this p oin t new prop erties ha ve become more difficult to prov e and feel more and more like research. 5 One is the Only T riangular Cub e I am particularly interested in n u m b ers th at sim ultaneously have tw o of my regular prop erties . F or example, I w ond ered if th ere were any pronic cub es. Since p ron ic numbers are of the form n ( n + 1) and cub es are m 3 , I needed to find p ositiv e solutions to the equ atio n n ( n + 1) = m 3 . As num b ers n an d n + 1 are coprime, each of th em m ust b e a cub e. The only tw o integer cub es that d iffer by 1 are 0 and 1. Hence, pronic cub es do not exist. Next I mov ed to triangular cu bes. Obviously , 1 is b oth t ria ngular and cu bic. I sp ent a lot of my computer’s processing p o wer try in g to fi nd other triangular cu bes. When I realized that 1 is t h e only triangular cub e I could find, I started to look around for the proof. I asked my friends on the OEI S [8] Seq F an mailing list if any of th em knew the proof. I receiv ed proofs fro m Jaap Sp ies and Max Alekseyev. Here’s Max’s pro of . T riangular n u mb ers are of the form n ( n + 1) / 2 and cub es are m 3 . Hence, triangular cub es correspond t o the solutions of the equ atio n n ( n + 1) / 2 = m 3 . The big tric k in solving this eq uation is to m ultiply b oth sides by 8: 4 n ( n + 1) = 8 m 3 . Then w e can rearrange the equation into: (2 n + 1) 2 − 1 = (2 m ) 3 . By th e Catalan’s conjecture, whic h was recently prov en, th e only tw o p ositiv e p o wers that differ by 1 are 9 and 8. F rom here the pro of follo ws. 6 Odd Fib onaccis I stumbled up on the follo wing statement in the MathW orld [9]: n o o dd Fib onacci is divisible by 17. S o I started w ond ering if I can make a unique property out of it. Is 17 th e only such n umber? Clearl y not, as multiples of 17 will also hav e the same prop ert y . Is 17 the smallest such num b er? Obviously , no o dd Fib onacci is divisible by 2. So 17 is not the smallest. What if we exclud e 2? By think ing it through, I proved tw o new u nique prop erties for my website and wrote a pap er on the w ay [5]. The tw o prop erties are: 9 is the smallest o d d num b er such that no odd Fib onacci num b er is d ivisi ble by it. 17 is the smallest od d prime suc h that no o dd Fibonacci number is divisible by it. 3 7 Conjectures Here is a sample of my unique prop erties th at are conjectures: 70 is the largest known number n such that 2 n has a digit sum of n . 86 is conjectured to b e the larges t number n such that 2 n does not contain a 0. 264 is the largest known number whose square is u ndulating: 264 2 = 6969 6. 2201 is the only non-palindrome known to hav e a palindromic cub e. 3375 is the largest known cube which contai n s all prime digits. 8 Help Me Chec k Th ese Prop erties Y ou can help me with some prop erties. It is not enough t o send me a link to a website. I wan t either a proof or a program, p ref erably in a programming language I can read (Ja va , Mathematica) or a reference to a pap er in a refereed journal. I hav e 674 prop erties in my database th at n eed chec k ing. Here are some: • The maximum number of squares a chess bishop can visit, if it is on ly allow ed to visit each square once. • The num b er of legal knight/king/bishop mo ves in c h ess. • The smallest prime n u m b er that is th e sum of a prime num b er of consecutive prime n umb ers in a prime num b er of differen t wa ys. • The num b er of forms of magic knight’s tour on the c h ess b oard. • The num b er of primes that can app ear on a 24-hour and/or 12-hour d igi t al clock. • The smallest num b er t hat cannot b e added to a nonzero palindrome suc h that the sum is also palindromic. 9 Submissions If you submit a p roperty , I do not guarantee th at I will add it b ecause I already hav e ab out 1,500 properties of num b ers in my database that are not true, not unique or not very in t eres ting. In addition, I do not like prop erties that contain parameters. Parameters mean that there is a sequen ce and therefore the prop ert y is n ot unique. As a rule of thumb if you can write your prop ert y without using num b ers, I like it. Also, sometimes I allo w 2, 3, 10 or 100 as parameters. I n general, the bigger th e n u m b er the more accepting and forgiving I am. Also, if yo u hav e a pro of, please send the proof, too. If I like your prop ert y and un d erstand your p roof, I will upload the prop ert y v ery fas t and include your name as a sub mitter. If I do not un derstand th e pro of for your prop ert y , I will still hav e your p roperty and your name in my in t ernal database, but you will h a ve to wait u n til I get around to proving it. 10 Alexey Radul My son, Alexey Radul, suggested the idea for Number Gossip abou t 10 y ears ago and at first I u ploaded it on my personal websi te [6 ]. In 2006 Alexey redesigned Number Gossip using Ruby on Rails an d at the end of 2006 it got its own url http://www.n umbergossip.com/ . Alexey wa s also one of th e first submitters. Here is one of his submissions: 6 is the only even evil p erfect num b er. 4 Let me remind yo u th at p erfect numbers are num b ers that are sums of their prop er d iv is ors. F or example: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. It is not known if o dd p erfect num b ers exist. But for even p erfect num b ers it is known th at they are of the form: 2 p − 1 (2 p 1) for prime p . Actually , we kn ow even more than t hat: we kno w that p erfect numbers are in a one-to-one correspond ence with p , for which 2 p − 1 is prime. That means, that even p erfect num b ers are in a one-to-one correspond ence with Mersenne primes. If a num b er is represen ted with the p o wer s of t wo, u sual ly it is easy to find its binary representati on. F rom here you can pro ve that the evilness of 6 follo ws from the ev en ness of 2. The only even prime is 2, hence t he only evil even p erfect num b er is 6. 11 Statistics Currently , the highest number you can input on Nu m b er Gossip is 9,999. Other than th e sp ecial case of number 1, every number has at least four prop erties, one from eac h of the follow in g groups: • even or o dd • prime or comp osite • evil or o dious • p erfe ct, abun dan t or deficient During m y talk I ann ounced that the smallest num b er that d oes not hav e a unique prop ert y is 32. S omeo ne p oin ted out to me th at this fact is in itself a uniqu e prop ert y . I do not w ant to add self-referencing p roperties to my database, but on my wa y home from G4G8 I inv ented a p roperty for 32: 32 is conjectured to b e the highest p o we r of tw o with all prime digits. I c heck ed this prop erty u p t o 86 th p o wer of 2. And 86 is conjectured to be th e highest pow er of tw o that doesn’t contai n zero. That means that 32 conjecture follo ws from 86 conjecture. The next num b er without a unique property is 51, follow ed b y 56 and 57. The largest num b er that has a unique prop ert y is currently 8833: 8833 is the largest 4-digit number that is the sum of the sq u ares of its h alves: 8833 = 88 2 + 33 2 . 12 F uture Plans I’ve got really big p lans for Number Gossip: • I hav e 647 more unique prop erties in m y in tern al database t hat I need t o c h ec k and I am alw ays discov ering new prop erties. • I ha ve a list of about ten more regular properties I w ant to add, lik e brillian t, fortunate, primev al and totient num b ers. • I would lik e to mov e the limit from 9,999 up to 20,000. • As I p rog ress, the n ew prop erties are increasingly harder to p ro ve. I w ould like to pro v ide pro of s of difficult prop erties on the w ebsite b ecause I th ink you w ould find that interesting. • Some of m y unique prop erties are of the form “th e smallest n u mb er that . . . ” or “the largest num b er th at . . . ” These prop erties can sometimes have a corresp onding sequence in the online database (see OEI S [8]). When they do, I will b e adding references to the O EIS. 5 13 Ac kn o wle dgmen ts I am lo oking for unique properties everywhere I can. I got m y initial encouragement from Erich F riedman’s page “What’s Sp ecial Ab out This Number?” (see [2]). Later I used the Online Encyclop edia of Integer Sequences [8], Wikip edia [10] and the “Prime Curios!” collection (see [7]). I am grateful to Su e Katz for helping me to edit this pap er. References [1] Elwyn R. Berlek amp, John H. Conw ay , R ic hard K. Guy , Winn ing W a ys for Y our Mathematical Plays, V. 3, p. 463-464, 2003. [2] Eric h F rie dman’s page “What’s Sp ecial Ab out This Number?”, published at http://www.stets on.e du/ ∼ efrie dma/numb ers.html [3] Martin Gardner, Mathematical Circus, p . 128, p. 135, 1979. [4] T an ya Kh o v anov a, Autobiographical Numbers, arXiv:0803.027 0 v1 at http://arxiv.or g/abs /0803.0270 [5] T an ya Kh o v anov a, 9 Divides no Odd Fib onacci, arXiv:0712. 3509 v1 at http://arxiv.or g/abs/0712.3509 [6] T an ya Kh o v anov a, personal webpage at http://www.tanyakhovan ova.c om [7] G. L. Honaker, Jr., con tent editor, Prime Curios! col lection at http://primes.utm.e du/curios/ [8] N. J. A. Sloane, Online Encyclop edia of Integer Sequences (OEIS), p u blished electronically at http://www.r ese ar ch.att.c om/ ∼ njas/se quen c es/ [9] Eric W. W eisstein, “Fibonacci Numbers.” F rom MathW o rld – A W olfra m W eb Resource. http://mathworld.wolfr am.c om/Fib onac ciNumb er.html [10] Wikip edia on I ntegers at http://en.wikip e dia.or g/wiki /Cat e gory:I nt e gers 6