Problems & Puzzles: Puzzles

 Puzzle 803. Panprimmatics An article from Greg Ross recently recalled my attention about pangrams: "is a sentence using every letter of the alphabet at least once (see this). An example in English is  "The quick brown fox jumps over a lazy dog." What about some equivalence for prime numbers, let's call them "a panprimmatic"? Well, we need impose some restrictions because natural alphabets are finite and primes are infinite. So, let's ask for numbers that contain in his decimal expression the first 25 prime numbers (from 2 to 97, avoiding permitting that digits may be used simultaneously for two distinct prime numbers (for example the digit "1" in the decimal string "...617..."). I'll define this number a "panprimmatic2-97" I'll give you two examples found by me this week: 1) pi up to 1000 digits, contains a string (in bold below) of 197 decimal digits which contains all of the first 25 prime numbers pi=3.141592653589793238462643383279502884197169399375105820974944592307816406286208 99862803482534211706798214808651328230664709384460955058223172535940812848111745 02841027019385211055596446229489549303819644288109756659334461284756482337867831 65271201909145648566923460348610454326648213393607260249141273724587006606315588 17488152092096282925409171536436789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185480744623799627495673518857527 24891227938183011949129833673362440656643086021394946395224737190702179860943702 77053921717629317675238467481846766940513200056812714526356082778577134275778960 91736371787214684409012249534301465495853710507922796892589235420199561121290219 60864034418159813629774771309960518707211349999998372978049951059731732816096318 59502445945534690830264252230825334468503526193118817101000313783875288658753320 83814206171776691473035982534904287554687311595628638823537875937519577818577805 321712268066130019278766111959092164201989 2) sqrt(379) up to 1000 digits contains a string of 88 digits which contains all of the first 25 prime numbers: sqrt(379) = ...4153114356837073089364034797853167107135059185868117082 590178726980193502303929385808561... BTW, I looked up to 1000 decimal digits just due to my programming limitations. You are able to search as far as you can. The asked solutions are not "ad-hoc or artificial numbers like the concatenation of the first 25 prime numbers, or things like that. The puzzle intends you to analyze the decimal expressions of numbers like x^y, x/y, sqrt(x), x!, ... or mathematical constants like pi, e, gamma, ... or universal physical constants, and so on. Q1. Find your best non-adhoc number that contains a string as short as possible satisfying the asked condition. Q2. Perhaps you devise a more interesting way of define a panprimmatic. Yes?... Tell us, and send your best solution!

Contributions came from Jan van Delden, Dmitry Kamenetsky, Carlos Rivera and Emmanuel Vantieghem.

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Jan wrote:

Suppose the shortest decimal expansion containing the first 25 primes is equal to 0.a[1]..a[n]. An answer would then be a[1]..a[n]*10^n/10^(n+1). If the decimals repeat, like x=0.a[1]..a[n]a[1]..a[n] etc. we could find a fraction representation by solving x*10^n – a[1]..a[n]=x. So x=a[1]..a[n]/(10^n-1).
A fraction can always be found, or in other words: I couldn’t find the time to search for a non-rational solution.

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Dmitry wrote:

This is not the answer you were looking for, but I think it may interest you. The following is the smallest number I found that contains the first 25 primes as substrings: 23112941343471735359619678378979. This number has 32 digits. There is a sequence for such numbers: https://oeis.org/A054261

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Carlos Rivera wrote:

I look for integers of the type p^x, and found a string-solution of 58 digits inside the decimal expression of 73^928. Here is it:

73^928 = 1457518888971825146008743390808314252362265482075739167
80964941191257714468391039344452644484654662419396430826498790567173043112221185
42484299508665499322079692960820727242259476630225956043856074280802592994410952
17367914137536715296223195948791736537161195897499588083824614347326291751851543
24893098797338458945847619720711952457208146730828365630982946458568463126157228
56453961843654425209840706677018450350025428977971583104407461434520485629453865
83436909493504426018307140205611562590473437745481743200535707941919044986829173
08320911016434986043555933760205972498322047351674282352361647768170655010448174
45308203526005292876013600661670126048269205561741876090888589970129033645669982
04198680089051109994794565501049494256916857677152138419500785558719362495504669
22381771004164542454292504213857823722123281081879059364410601705892481502045637
56902948597911524088009407452155154543809822367954854949162765349195244255505951
64881206969745831251731923630702267869146356820292004382345557592269572881922377
56592235707213665073261908238994021217711135505218782777275968947142822709047135
89141815250008471878338287932555820891054170388639135558125409897918622562772946
31557991477376172383009158979906510404460787855798698959064134968998860518270646
92736984315189004781667924126408601947355063418016512221255460267426603615974439
19779986838883269203797377887656187166813613052469404198131265967786764577431808
18518835309588950477764099064959025466227076224191557801885548131801571194418978
52299457197235865292665396050919640536119526955432492094947309664271496665391169
44773990766510263543590427415755861617733251836381127501629643994637285118264228
928221976603394566229910779754236247529747725947059143554041793173864321281
(1730 digits)

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Emmanuel wrote:

Here is the smallest panprimatic I found in the decimal expansions of  e (the base of the natural logarithms) :
471199783441539679308089243892118069783103014170732659237462931368061 (69 digits). It occurs at the  8978601 th position after the decimal point. If there is a smaller one (which is highly probable) it must be after the twenty milionth decimal.

I examined the numbers  Pi, Gamma, Sqrt(2) and the golden ratio (sqrt5 -1)/2 upto 20000000 digits and there was no smaller panprimatic.

About squares, in my opinion it is very difficult to find the least square that contains a panprimatic.
My best result is 20536517191858768239897995424582451^2 =
421748538371510747925237473136611198937434135293718594441747697167401 (69 digits).  It contains the 60-digit panprimatic: 538371510747925237473136611198937434135293718594441747697167.

I know of a 67-digt square ( 2280030103071332353057552679172331^2 ) but it contains a 63-digit panprimatic.

Later, on Oct 16, 2015, Emmanuel wrote again this:

The square number  11317197372327489831615994439129479567853641 = 3364104245163560598371^2  has 44 digits and is panprimatic.. I'm convinced that there exist smaller panprimatic squares but that they are very hard to find.

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