3
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Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$

The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are :

? for(j=5,2000,p=prime(j);x=eval(Vec(Str(p)));l=length(x);print1(x[l-1]))
11112233444556677788900001233345566778999912223345566778890111334455677889001233
34456667899002244566778990011134445567789001233455667890122223555678880112344567
78990112333456668999001225567889011223345778899000122667890222334555788889912344
55677890001122356669990223445578880123466777780013345778999011223566888891123344
56700123345667889901334455778889911234456770233445579901234556778889901112344567
89900133455678900123556679011234466788012366688900122555579001122344567789013456
66699112233445578890112334577799001233666799022345567880111223446800011224557799
91223355570111234455677889233456799022445568890111244668990233445567790222355888
99011367780013334556678990012235578890011456778902333677890023458899011133444777
80001223566789234455556889011344478900122344556667890223588012344567789012334567
99011224566778901122345566778922456789224556677890135567789000133667899022234566
78901144566778990112345670022255789011123445890023345691135557888901223444567788
90023446788990122455589122456777800123345690113556888901124667790123334667899911
25667881223446601223346789902224466788990113344567800123334666892234566790011244
45690023355678890022345788911234447799011233336667799123345801122344677891233446
68890112335557880022344670033666799901334556678912445567780012333456992235556789
01235689001233556689012233578893345566889003345778902455678891113456777913455677
89112255689912344678899012457899123578890113477880012233366890022334556788013444
77900011445669012345566788022447779001233557789901123445678801113445578901234568
99022245899011124556778000033446990022455677881124456990123336899112455667891235
67991123467889990122255568890234577780001233669901235578880445577902445890222446
88001123344678013344556699222335568911233445567789122345667899223455681135677890
02333467899123456677889011234567789012345667991245568800124444667782333346678990
01257888001123577900356666789901234888912234567013346668112234557889124566700133
4556799902445681223477880022334678891222344579901235688900033599912234557888

I calculated the continued fraction and the minimal-polynomials for $x$ with PARI/GP and the number seems to be transcendental.

  • Is x rational, irrational algebraic or transcendental ?
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  • 2
    $\begingroup$ That question is difficult to answer and most likely unsolved. Much remains unknown about transcendental numbers. $\endgroup$ – Aleksandar Jun 14 '15 at 21:06
  • $\begingroup$ Perhaps, at least the irrationality can be proven. $\endgroup$ – Peter Jun 14 '15 at 21:07
  • $\begingroup$ Irrationality is also a grey area in mathematics, but perhaps. $\endgroup$ – Aleksandar Jun 14 '15 at 21:09
  • $\begingroup$ "Of yourse" this number is transcendental, but how to prove that? At least if it were rational then there'd exist $k\ge 1$ such that $p_{m+k}\equiv p_m\pmod {100}$ for almost all $m$. But it seems plausible that one can show someting like: There are arbitrarily long sequences of consecutive primes such thatnone of them is $\equiv 1\pmod{100}$. $\endgroup$ – Hagen von Eitzen Jun 14 '15 at 21:16

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