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.

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.

My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains #ifdefs 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.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):

This is equivalent to
echo 41 | ./27

Output:

(please click 'Go !')

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)

Benchmark

The correct solution to the original Project Euler problem was found in 0.07 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

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 never 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:

Python: www.mathblog.dk/project-euler-27-quadratic-formula-primes-consecutive-values/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p027.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p027.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/20-29/problem27.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem027.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/27 Quadratic primes.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler027.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.

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

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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !