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Thread: 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 $\displaystyle 1257787log(2)=378631.615156$

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

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

    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 $\displaystyle 2^{1257787}-1$ are the same as those of $\displaystyle 2^{1257787}$, so we need only consider the latter.

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

    Now the $\displaystyle 378631$ 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 $\displaystyle 0.615156$ tells you what the digits are.

    RonL
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