How many decimal digits does the number $2^{2048}$ have, and why does this matter for RSA?

$2^{2048}$ has 617 decimal digits — a number vastly larger than the number of atoms in the universe ($10^{80}$).

Approximation formulas:

• $2^{10} = 1024 \approx 10^3$, so $2^{10} \approx 10^3$
• Number of digits of $2^x$: $\lfloor x \cdot \log_{10} 2 \rfloor + 1 \approx x \cdot 0.3$

For $2^{2048}$:

• Shortcut: $2048 \times 0.3 = 614.4$, so roughly 614 digits
• Exact: $2048 \times \log_{10} 2 = 2048 \times 0.30103 = 616.5$, so $2^{2048} \approx 3.2 \times 10^{616}$, which has 617 digits
• The shortcut undershoots by about 2 digits because it rounds $\log_{10} 2 = 0.30103$ down to 0.3. Good enough for an order of magnitude, but the error grows with the exponent

Why it matters: RSA uses a modulus $N = p \cdot q$ of this size. All arithmetic is done with these ~600-digit numbers. The key insight is that even though these numbers are enormous, the SAM algorithm and modular reduction keep computations feasible — while the inverse operations (factoring, discrete roots) remain infeasible.

Scale comparison:

• Atoms in the Milky Way: ~$10^{67}$
• Atoms in the observable universe: ~$10^{80}$
• $2^{256}$ (AES key space): ~$10^{77}$
• $2^{2048}$ (RSA modulus): ~$10^{616}$ — incomprehensibly larger

Go deeper:

• RSA (cryptosystem) (Wikipedia) — where these ~600-digit moduli actually get used.
• Encryption and HUGE numbers (Numberphile) — an intuition for just how large these numbers are.