Number of Digits in Power of 2/Examples/Mersenne Number M127
Jump to navigation
Jump to search
Example of Number of Digits in Power of 2
When expressed in conventional decimal notation, the number of digits in the Mersenne number $M_{127}$ is $39$.
Proof
Let $m$ be the number of digits in the Mersenne number $M_{127}$.
Recall the definition Mersenne number $M_{127}$:
- $M_{127} = 2^{127} - 1$
We have that $2^{127}$ is not a power of $10$.
Neither can $2^{127} - 1$ be a power of $10$.
So $M_{127}$ and $2^{127}$ have the same number of digits.
From Number of Digits in Power of 2:
- $m = \ceiling {127 \log_{10} 2}$
From Common Logarithm of 2:
- $\log_{10} 2 \approx 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
and so:
- $m = \ceiling {38 \cdotp 23}$
Hence the result.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 301 \, 029 \, 995 \, 663 \, 981 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 30102 \, 99956 \, 63981 \ldots$