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A089578
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Decimal expansion of 2^20996011 - 1, the 40th Mersenne prime A000668(40).
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4
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1, 2, 5, 9, 7, 6, 8, 9, 5, 4, 5, 0, 3, 3, 0, 1, 0, 5, 0, 2, 0, 4, 9, 4, 3, 0, 9, 5, 7, 4, 8, 2, 4, 3, 1, 1, 4, 5, 5, 9, 9, 3, 4, 1, 6, 0, 8, 5, 3, 5, 1, 8, 3, 5, 9, 5, 2, 2, 5, 4, 6, 7, 0, 1, 2, 5, 6, 5, 4, 9, 8, 7, 6, 8, 9, 0, 8, 3, 5, 1, 5, 6, 0, 2, 2, 1, 2, 4, 0, 0, 9, 6, 8, 0, 2, 8, 2, 8, 5, 3, 6, 1, 3, 2, 5
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OFFSET
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6320430,2
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COMMENTS
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We can compute the digits of 2^p directly by noting that 2^p = 10^(p*log(2)/log(10)) = 10^(p*log_10(2)). This result is 10^(i+f) where i is the integer part and f the fractional part. Then 10^f will produce a decimal number i.d1d2d3d4... where i is an integer from 1 to 9 (zero cannot occur in i) and d1, d2 ... are the digits in the fractional part where 0 is allowed. So i is the first digit in 2^p, d1 the second, d2 the third etc. The expansion is self evident in the PARI program. This routine allows the direct computation of the digits of any base to a power: k^p = 10^(p*log_10(k)).
The 40th Mersenne prime found by GIMPS / Michael Shafer in 2003 is 1259768954503301...4065762855682047 = 2^20996011 - 1. The second PARI program below computes all digits. - Georg Fischer, Mar 18 2019
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LINKS
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MATHEMATICA
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RealDigits[10^N[20996011Log[10, 2] - 6320430, 105]][[1]] (* Georg Fischer, Mar 19 2019 after Jakob Vecht in A117853 *)
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PROG
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(PARI) \\ digits of the 40th Mersenne prime: 2^20996011 - 1
p = 20996011; digitsm40(n, p) = { default(realprecision, n); p10 = frac(p*log(2)/log(10)); v = 10^p10; for(j=1, n, d=floor(v); v=frac(v)*10; print1(d", ") ) }
digitsm40(105, p)
(PARI) write("a089578.txt", 2^20996011 - 1) \\ Georg Fischer, Mar 18 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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