#99 - Largest exponential
Comparing the two numbres written in index form like $2^{11}$ and $3^7$ is not difficult, as any calculator would confirm that $2^{11}=2048<3^7=2187$.
However, confirming that $632382^{518061}>519432^{525806}$ would be much more difficult, as both numbers contain over three million digits.
Using base_exp.txt (right click and ‘Save Link/Target As…’), a 22K text file containing one thousand lines with a base/exponent pair on each line, determine which line number has the greatest numerical value.
The first two lines in the file represent the numbers in the example given above.
Even calculating all these large numbers in Python is not that efficient. We can take the logarithm of both sides, and retain the monotonicity of the exponentials. So if beforehand, $a^b > c^d$, then $\log a^b > \log c^d$. We can simplify this further with $b\log a > d\log c$, which will be the condition we’ll check.
# file: "problem099.py"
exponents = np.genfromtxt(
'p099_base_exp.txt', delimiter=',', dtype=np.int64)
logs = exponents[:, 1] * np.log10(exponents[:, 0])
# Now find the maximum argument and print its line
index = np.argmax(logs)
print(index + 1)
print(exponents[index].tolist())
Running this short code results in an output,
709
[895447 504922]
0.011759600000000037 seconds.
Thus, $895447^{504922}$ is the largest number in our list, and it happens at line 709.
