<< 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.
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 toecho "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)
My code
… was written in C++11 and can be compiled with G++. You can download it, too. Or just jump to my GitHub repository.
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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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:
C# www.mathblog.dk/project-euler-97-digits-non-mersenne-prime/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/097.cs (written by Haochen Liu)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-097.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p097.py (written by Nayuki)
Python github.com/sefakilic/euler/blob/master/python/euler097.py (written by Sefa Kilic)
Java github.com/dcrousso/ProjectEuler/blob/master/PE097.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p097.java (written by Nayuki)
Go github.com/frrad/project-euler/blob/master/golang/Problem097.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p097.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p097.hs (written by Nayuki)
Haskell github.com/roosephu/project-euler/blob/master/97.hs (written by Yuping Luo)
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_097.clj (written by Guillaume Nargeot)
Clojure github.com/rm-hull/project-euler/blob/master/src/euler097.clj (written by Richard Hull)
Scala github.com/samskivert/euler-scala/blob/master/Euler097.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/097-Large-non-Mersenne-prime.pl (written by Gustaf Erikson)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p097.rs
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
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 >> |