<< problem 122 - Efficient exponentiation | Ordered radicals - problem 124 >> |
Problem 123: Prime square remainders
(see projecteuler.net/problem=123)
Let p_n be the nth prime: 2, 3, 5, 7, 11, ..., and let r be the remainder when (p_n-1)^n + (p_n+1)^n is divided by pn^2.
For example, when n = 3, p^3 = 5, and 4^3 + 6^3 = 280 == 5 mod 25.
The least value of n for which the remainder first exceeds 10^9 is 7037.
Find the least value of n for which the remainder first exceeds 10^10.
My Algorithm
This is basically problem 120 merged with a prime sieve.
My algorithm description of problem 120 explains why the maximum remainder is 2na (and a is a prime in the current problem).
Note
There is no need to check the first 7036 prime numbers because the problem statement mentioned that all of them produce a remainder <10^9.
But the computation is so simple and fast that it doesn't affect the execution time at all.
I prefer a prime sieve based on trial division because it can be computed incrementally without knowing an upper limit beforehand
(which is a must for the sieve of Eratosthenes).
Interactive test
This feature is not available for the current problem.
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
#include <vector>
int main()
{
// sieve based on trial division
std::vector<unsigned int> primes = { 2 };
for (unsigned int i = 3; ; i += 2)
{
bool isPrime = true;
// test against all prime numbers we have so far (in ascending order)
for (auto p : primes)
{
// next prime is too large to be a divisor ?
if (p*p > i)
break;
// divisible ? => not prime
if (i % p == 0)
{
isPrime = false;
break;
}
}
// no prime ?
if (!isPrime)
continue;
primes.push_back(i);
// only every "odd" prime
if (primes.size() % 2 == 0)
continue;
// compute remainder
auto remainder = 2 * i * primes.size();
// exceeds 10^10 ?
if (remainder >= 10000000000ULL)
break;
}
// print result
std::cout << primes.size() << std::endl;
return 0;
}
This solution contains 8 empty lines, 9 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.02 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
May 24, 2017 submitted solution
May 24, 2017 added comments
Difficulty
Project Euler ranks this problem at 30% (out of 100%).
Links
projecteuler.net/thread=123 - 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-123-remainder/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/123.cs (written by Haochen Liu)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-123.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p123.py (written by Nayuki)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/123.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/123.cpp (written by Yuping Luo)
C github.com/LaurentMazare/ProjectEuler/blob/master/e123.c (written by Laurent Mazare)
Java github.com/dcrousso/ProjectEuler/blob/master/PE123.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p123.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem123.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem123.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p123.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/123.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p123.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler123.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/123-Prime-square-remainders.pl (written by Gustaf Erikson)
Perl github.com/shlomif/project-euler/blob/master/project-euler/123/euler-123.pl (written by Shlomi Fish)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p123.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 122 - Efficient exponentiation | Ordered radicals - problem 124 >> |