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

# 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.

# My 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)
{
// fast exponentiation
if (exponent & 1)
result = (result * base) % modulo;
base = (base * base) % modulo;
exponent >>= 1;
}
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 9 empty lines, 8 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 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 rarely 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 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 12,983 points (out of 15100 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler. Thanks for all their endless effort.

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