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Math Help - Finding digits of a prime

  1. #1
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    Finding digits of a prime

    I'd like to know how to find the first 3 digits and the number of digits of the prime number 2^1257787-1. I think I'm supposed to use logarithms.
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  2. #2
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    To find the number of digits, use 1257787log(2)=378631.615156

    There are 378632 digits in the number, not subtracting the 1.

    For the first few digits of the huge number, use 10^{.615156}=4.12217....

    The first 3 digits are 412.
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  3. #3
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    How does the log of the base times the exponent equal the number of digits for b^n -1 ?
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    bump

    Ok, I understand there are 378632 digits, but why can we find the first few digits by using 10^.6151562...
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  5. #5
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    Quote Originally Posted by frankdent1 View Post
    Ok, I understand there are 378632 digits, but why can we find the first few digits by using 10^.6151562...
    First the leading 3 digits of 2^{1257787}-1 are the same as those of 2^{1257787}, so we need only consider the latter.

    Now: 2^{1257787}= 10^{\log_{10}(2)\times 1257787}\approx 10^{378631.615156}.

    Now the 378631 in the exponent above just tells you how far the decimal point in the answer is to the left of the most significant digit, and the fractional part 0.615156 tells you what the digits are.

    RonL
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