<< problem 96 - Su Doku Anagramic squares - problem 98 >>

# Problem 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 * 2^7830457 + 1.

Find the last ten digits of this prime number.

# Algorithm

I used powmod in problem 48 to compute a^b \mod c (see there for an explanation of the way powmod works).
Unfortunately we have to find the 10 last digits, which is one digit too much for 32/64 bit multiplications.
GCC's support for 64/128 arithmetic solves this problem easily.

## Alternative Approaches

I could have used my old powmod code which relies on mulmod but that code is too slow for the Hackerrank version of this problem.
On the contrary, that code from problem 48 is more portable.

# My code

… was written in C++11 and can be compiled with G++. You can download it, too.

The code contains #ifdefs to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL to produce the result for the original problem (default setting for most problems).

       #include <iostream>
#include <iomanip>

// GCC only !!!
typedef unsigned __int128 BigNum;

//#define ORIGINAL
#ifdef ORIGINAL
const unsigned int Digits = 10;
const BigNum       Modulo = 10000000000ULL;
#else
const unsigned int Digits = 12;
const BigNum       Modulo = 1000000000000ULL;
#endif

// compute the n-th power of a big number (n >= 0)
BigNum powmod(BigNum base, unsigned int exponent, BigNum modulo)
{
BigNum result = 1;
while (exponent > 0)
{
if ((exponent & 1) == 0)
{
// fast exponentiation
base = (base*base) % modulo;
exponent >>= 1;
}
else
{
// slower standard approach
exponent--;
result = (result * base) % modulo;
}
}
return result;
}

int main()
{
unsigned long long sum = 0;

unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
// read a * b^c + d
unsigned long long factor, base, exponent, add;
std::cin >> factor >> base >> exponent >> add;

// compute result
unsigned long long result = (powmod(base, exponent, Modulo) * factor + add) % Modulo;

// modulo all the way ... we need only the last 10 (or 12) digits
sum += result;
sum %= Modulo;
}

std::cout << std::setfill('0') << std::setw(Digits) << sum;
return 0;
}


This solution contains 8 empty lines, 9 comments and 5 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This live test is based on the Hackerrank problem.

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 2 3 4 5" | ./97

Output:

Note: the original problem's input 28433 2 7830457 1 cannot be entered
because just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

March 2, 2017 submitted solution

# Hackerrank

My code solves 11 out of 11 test cases (score: 100%)

# Difficulty

Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as hard.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

projecteuler.net/thread=97 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-97-digits-non-mersenne-prime/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p097.java (written by Nayuki)
Scala: github.com/samskivert/euler-scala/blob/master/Euler097.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
 << problem 96 - Su Doku Anagramic squares - problem 98 >>
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