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# 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 `#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.

# Interactive test

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

This is equivalent to`echo 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)*

# 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 solved **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 already 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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<< problem 26 - Reciprocal cycles | Number spiral diagonals - problem 28 >> |