[math-fun] New Year's oddity
Lots of things will change with the new year in a few days. But there's one thing that won't: The power of two in which the year first appears. As we all know, 2^212 is 6582018229284824168619876730229402019930943462534319453394436096. What you may not have noticed, not being as perceptive as me (i.e. having a life) is that it contains both 2018 and 2019 as substrings. And that it's the *first* power of two to contain *either* of those numbers as substrings. When, if ever, is the last time this happened? When, if ever, will be the next time? What's the expected density of this sequence? And should I add it to OEIS? Of course the question can be extended to other radices than base ten, and to other powers than those of two. A problem I haven't solved is which is the first power of two to start with 2018, and which is the first to start with 2019. It's easy to prove that there must be a solution, not just for 2018 and 2019, but for any positive integer.
Heuristically there should be infinitely many numbers with this property, and they should be about as common as square numbers (to within a constant multiplicative factor). In particular, let S be the string: "1248163264128256512[...]" obtained by concatenating all of the powers of 2. For a given integer n, let f(n) be the position of the first occurrence of the decimal representation of n as a substring of S. If we instead use a random string of uniform iid digits in place of S, then f(n) will follow a geometric distribution with mean 10^len(str(n)), which lies between n and 10 n. If we assume independence of f(n) and f(n+1) (i.e. we generate a different random string each time), then we can find the probability that f(n) and f(n+1) belong to the same power of 2; it is not too difficult to show that this is asymptotically proportional to 1/sqrt(n). But of course this is just a heuristic argument rather than a mathematical proof; finding the latter is incredibly difficult. Best wishes, Adam P. Goucher
Sent: Sunday, December 23, 2018 at 4:14 PM From: "Keith F. Lynch" <kfl@KeithLynch.net> To: math-fun@mailman.xmission.com Subject: [math-fun] New Year's oddity
Lots of things will change with the new year in a few days. But there's one thing that won't: The power of two in which the year first appears.
As we all know, 2^212 is 6582018229284824168619876730229402019930943462534319453394436096. What you may not have noticed, not being as perceptive as me (i.e. having a life) is that it contains both 2018 and 2019 as substrings. And that it's the *first* power of two to contain *either* of those numbers as substrings.
When, if ever, is the last time this happened? When, if ever, will be the next time? What's the expected density of this sequence? And should I add it to OEIS?
Of course the question can be extended to other radices than base ten, and to other powers than those of two.
A problem I haven't solved is which is the first power of two to start with 2018, and which is the first to start with 2019. It's easy to prove that there must be a solution, not just for 2018 and 2019, but for any positive integer.
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A little bit of python-fu shows that the first occurrences you were looking for are:
2**4665
2018040010421574775571604060422643201505689312119712695770311734234926031950651023616765393358044185207801932457776427695202228030110264613880493874514140718332599004080282038518019804999192077130149201787530517094251916544978883848317206969576889770902168182899773611396021314744100217186765810503450860221447352681927738900492886954312591224226451835384848602472763197392579973895225861290264585024395883423413402124958878436820275807426759126677685845878483935711415045142996679282444666483864809422690244312845986948889520962516307001550006704902191084277142093022451100323301621779658810702350078457173828268598795728883348371728205548407937946952522996947012061238210856095825529440314947923568809071518183034243353319130600667864621373043221202383519293612744898552110334463357381801321180257133908561655793761373061399283931413412042152752123446284905149396492021638512230742155533622066126639992434626235604976868135470912550837175421937629715257730171046927099236158084731668835303902010693823390700967405964955670755130911461484639361358114897004690518963133948113553483874902438939614960792839700149072720237183019451229525257218804560439616971637628056498334912637843629780600766554960353806764425987362432719662148642697969359077185190128395872668014815243417223782521150079665282234263681444068555325447309987783736972916390879762338466306561026627770328871397156975450453782011524763615232L
2**2044
2019812879456937956294679793041871997527756416857217752008146589220290946179243180824825088220182091480544872557618626218382472446905682568443009524153017695039429835456312255734387359399353256674753602399004223017299513665163734760114880896154760654411352865752269065180473493221316613037972024945245649095119645836854271401292810924160285593428511002207895128629862853708189137044278769634391162054011069795371475232403866084849896947018852869025231100827080451951695355294426263107822318857933207716854908911291043620940374829272062414888470322899339833471475133464576850290332404912809509262097240850691224764416L
Some more scripting gives me this list of years where the phenomena has happened and the associated power of 2: 58 33 73 30 110 42 784 74 792 56 913 76 956 75 984 54 1069 172 1466 228 2018 212 2098 206 2331 135 2609 262 2933 229 3027 311 3082 133 3310 135 3333 219 3972 188 4197 207 4207 409 4333 182 4589 178 4688 183 4784 74 5246 204 5349 117 5534 171 6165 134 6334 210 6668 220 6760 318 7166 181 7339 217 7706 131 7979 152 8667 183 8989 151 9179 179 9839 307 /f On Sun, Dec 23, 2018 at 5:14 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Lots of things will change with the new year in a few days. But there's one thing that won't: The power of two in which the year first appears.
As we all know, 2^212 is 6582018229284824168619876730229402019930943462534319453394436096. What you may not have noticed, not being as perceptive as me (i.e. having a life) is that it contains both 2018 and 2019 as substrings. And that it's the *first* power of two to contain *either* of those numbers as substrings.
When, if ever, is the last time this happened? When, if ever, will be the next time? What's the expected density of this sequence? And should I add it to OEIS?
Of course the question can be extended to other radices than base ten, and to other powers than those of two.
A problem I haven't solved is which is the first power of two to start with 2018, and which is the first to start with 2019. It's easy to prove that there must be a solution, not just for 2018 and 2019, but for any positive integer.
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Keith, we do have the sequence 10, 0, 1, 5, 2, 8, 4, 15, 3, 12, 10, 40, 7, ..., which of course you will immediately recognize. It also has an easily remembered A-number, A030000 There is also the companion 0, 1, 5, 2, 9, 6, 46, 3, 53, 10, 50, 7, 17, 47, ..., A018856. On Sun, Dec 23, 2018 at 2:31 PM Dave Dyer <ddyer@real-me.net> wrote:
The line between math and numerology is pretty thin.
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participants (5)
-
Adam P. Goucher -
Dave Dyer -
Frank Stevenson -
Keith F. Lynch -
Neil Sloane