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A230517
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An irrational x such that the decimal representation of neither x nor sqrt(x) contains the digit 0.
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0
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1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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0,2
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COMMENTS
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The rational number 1/9 is an example of a number in [0, 1] such that the decimal representation of neither x nor sqrt(x) contains the digit 0. The object of Problem 10439 of the Amer. Math. Monthly was to find an irrational with the same property (see link).
The solution proposed by Jerrold Grossman defines a sequence of irrationals starting with c1= 0.121121112... (A042974). Moving from left to right, the 0's in the decimal expansion of sqrt(cn) are eliminated by increasing the corresponding digit in the decimal expansion of cn by 2. The limit of cn is a number with the desired property.
The indices of the decimals that are successively changed are 4, 8, 29, 38, 40, 54, 62, 70, 72, 96, 118, ... (see print(ndeci) in PARI script).
The decimal expansion of sqrt(x) begins with 0.3483118317127931144162557719319698175373163374567....
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LINKS
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EXAMPLE
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0.12132113211112111112111111213111112113131112111111111411111113...
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PROG
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(PARI) pdeci(x, nb) = {x = x * 10; for (n=1, nb, d = floor(x); x = (x-d)*10; print1(d, ", "); ); print(); }
finddeci(x) = {x = x * 10; found = 0; nd = 1; while (! found, d = floor(x); x = (x-d)*10; if (d == 0, found = 1, nd++); ); nd; }
changedeci(x, ndeci) = {deci = floor(x * 10^ndeci) - 10*floor(x * 10^(ndeci-1)); x += 2/10^ndeci; x; }
lista(nn) = {prec = 2*nn; default(realprecision, prec); x = 0; for (n=1, prec, x = 10*x + 1 + issquare(9+8*n); ); x /= 10^prec; ok = 0; while (! ok, y = sqrt(x); ndeci = finddeci(y); print1(ndeci, ", "); x = changedeci(x, ndeci); if (ndeci > nn, ok =1); ); print(); pdeci(x, nn); print("sqrt(x)=", sqrt(x)); } \\ Michel Marcus, Oct 22 2013
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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