#97 - Large non-Mersenne prime
The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form $2^{6972593}-1$; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form $2^p-1$, have been found which contain more digits.
However, in 2004 there was found a massive non-Mersenne prime which contains 2,357,207 digits: $28433\times2^{7830457}+1$.
Find the last ten digits of this prime number.
We take the mod $10^{10}$ to find the last ten digits of a number. Calculate the number directly with Python:
# file: "problem097.py"
print((28433 * 2 ** 7830457 + 1) % (10 ** 10)) # one line!!
We get an answer instantly,
8739992577
0.038738599978387356 seconds.
Therefore, the last ten digits are 8739992577.
Alternative method
If our language is unable to hold large numbers, then we will need to take an approach that was first demonstrated in #48 - Self-powers, where through the use of clever modular tricks, we calculate the value of
\(28433\times2^{7830457}+1\mod 10^{10}\) directly. Please see that problem for implementation details. We would first calculate $2^{7830457}\mod 10^{10}$ before multiplying and adding through.
