<< problem 26 - Reciprocal cycles | Number spiral diagonals - problem 28 >> |
Problem 27: Quadratic primes
(see projecteuler.net/problem=27)
Euler discovered the remarkable quadratic formula: n^2+n+41
It turns out that the formula will produce 40 primes for the consecutive integer values 0<=n<=39.
However, when n=40, 40^2+40+41=40(40+1)+41 is divisible by 41, and certainly when n=41, 41^2+41+41 is clearly divisible by 41.
The incredible formula n^2-79n+1601 was discovered, which produces 80 primes for the consecutive values 0<=n<=79.
The product of the coefficients, -79 and 1601, is -126479.
Considering quadratics of the form:
n^2 + a * n + b, where |a|<1000 and |b|<=1000 where |n| is the modulus/absolute value of n e.g. |11|=11 and |-4|=4
Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n=0.
My Algorithm
Nothing fancy: isPrime
determines whether its parameter is a prime number or not.
And then two nested loops check every combination of a
and b
.
Note
isPrime
can be optimized in various ways - but the basic algorithm is fast enough for the problem.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 41 | ./27
Output:
Note: the original problem's input 1000
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++ and can be compiled with G++, Clang++, Visual C++. 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>
// return true if x is prime
bool isPrime(int x)
{
// reject invalid input
if (x <= 1)
return false;
// process all potential divisors
for (int factor = 2; factor*factor <= x; factor++)
if (x % factor == 0)
return false;
// no such divisor found, it's a prime number
return true;
}
int main()
{
// upper and lower limit of the coefficients
int limit;
std::cin >> limit;
// make sure it's a positive number
if (limit < 0)
limit = -limit;
// keep track of best sequence:
// number of generated primes
unsigned int consecutive = 0;
// its coefficients
int bestA = 0;
int bestB = 0;
// simple brute-force approach
for (int a = -limit; a <= +limit; a++)
for (int b = -limit; b <= +limit; b++)
{
// count number of consecutive prime numbers
unsigned int length = 0;
while (isPrime(length * length + a * length + b))
length++;
// better than before ?
if (consecutive < length)
{
consecutive = length;
bestA = a;
bestB = b;
}
}
#define ORIGINAL
#ifdef ORIGINAL
// print a*b
std::cout << (bestA * bestB) << std::endl;
#else
// print best factors
std::cout << bestA << " " << bestB << std::endl;
#endif
return 0;
}
This solution contains 8 empty lines, 14 comments and 5 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.07 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
February 23, 2017 submitted solution
April 5, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler027
My code solves 4 out of 4 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (out of 100%).
Hackerrank describes this problem as easy.
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=27 - 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-27-quadratic-formula-primes-consecutive-values/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/20-29/problem27.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p027.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/27 Quadratic primes.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem027.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p027.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler027.scala (written by Michael Bayne)
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 | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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 26 - Reciprocal cycles | Number spiral diagonals - problem 28 >> |