# Euler Problem 27

Euler published the remarkable quadratic formula:

n&sup2; + n + 41

It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 40 2 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 41&sup2; + 41 + 41 is clearly divisible by 41.

Using computers, the incredible formula  n&sup2; 79n + 1601 was discovered, which produces 80 primes for the consecutive values n = 0 to 79. The product of the coefficients, 79 and 1601, is 126479.

n&sup2; + an + 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.

﻿Not too hard. All variables tried must be primes. Ugly code, but its damn fast.

import java.util.*;

public class Euler27
{
public static void main(String[] args)
{
System.out.print(“Problem 27:\n”);
Euler27 e = new Euler27();
System.out.print(“Answer = ” + e.Problem()+ “\n”);
}
public String Problem ()
{
ArrayList<Integer> P = new ArrayList<Integer>();
P = Gen(1000);
int maxA = 0, maxB = 0, maxN = 0;
for (int a: P)
{
for (int b: P)
{
int c = 0;
for (int n=0; n <= 1000; n++)
if (P.contains(n*n + a*n + b))
c++;
else
break;
if (c > maxN)
{
maxA = a;
maxB = b;
maxN = c;
}
}
for (int b: P)
{
int c = 0;
for (int n=0; n <= 1000; n++)
if (P.contains(n*n + a*n – b))
c++;
else
break;
if (c > maxN)
{
maxA = a;
maxB = -b;
maxN = c;
}
}
}
for (int a: P)
{
for (int b: P)
{
int c = 0;
for (int n=0; n <= 1000; n++)
if (P.contains(n*n – a*n + b))
c++;
else
break;
if (c > maxN)
{
maxA = -a;
maxB = b;
maxN = c;
}
}
for (int b: P)
{
int c = 0;
for (int n=0; n <= 1000; n++)
if (P.contains(n*n – a*n – b))
c++;
else
break;
if (c > maxN)
{
maxA = -a;
maxB = -b;
maxN = c;
}
}
}
return String.valueOf(“A:”+maxA+” B:”+maxB+” for N:0-“+(maxN-1)+”,
so Product = “+(maxA*maxB)+”\n”);
}

public ArrayList<Integer> Gen (int limit)
{
ArrayList<Integer> p = new ArrayList<Integer>();
int x = 2, y = 0;
while (x < limit)
{
if (x % 2 != 0 || x == 2)
{
for (y = 2; y <= x/2; y++)
if (x % y == 0)
break;
if (y > x/2)