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# Problem 97: Large non-Mersenne prime

(see projecteuler.net/problem=97)

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 `#ifdef`

s 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;
}
// print with leading zeros
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.

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

May 8, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler097

My code solved **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.

# Links

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 already 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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