Is there a way to find the first digits of a number?

elementary-number-theory modular-arithmetic number-theory

Is there a way to find the first digits of a number?

For example, the largest known prime is $2^{43,112,609}-1$, and I did sometime before a induction to find the first digit of a prime like that. But, is there a way to find the first digits of a number?

To find the last x digits is easy, just calculate it mod $10^x$, but we can do something about the first ones?


What you want is $10$ to the power the fractional part of $43,112,609 \log_{10}2\approx 0.50033$, then $10^.50033\approx 3.1646$ so the leading digits are $316.$ Wolfram Alphaconfirms $31647$

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