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: $28433 \times 2^{7830457} + 1$.
Find the last ten digits of this prime number.

To find the last ten digits of the prime number $28433 \times 2^{7830457} + 1$, we can use modular arithmetic.

Python provides the built-in function `pow(x, y, z)` that computes $x^y \mod z$ efficiently. We can use this function to avoid calculating the whole number and focus only on the last ten digits.

Here is the Python code to calculate the last ten digits of the prime number:

“`python
base = 28433 # Base number
exponent = 7830457 # Exponent
modulus = 10**10 # Modulus for last ten digits

result = (base * pow(2, exponent, modulus) + 1) % modulus

print(“Last ten digits:”, result)
“`

Running this code will give you the output: `8739992577`, which are the last ten digits of the prime number $28433 \times 2^{7830457} + 1$.

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