#### Problem 27: Quadratic Primes

Euler discovered the remarkable quadratic formula: ** n^{2} + 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

^{2}+ 41 + 41 is clearly divisible by 41.

The incredible formula

**was discovered, which produces 80 primes for the consecutive values**

*n*- 79^{2}*n*+ 1601*n*= 0 to 79. The product of the coefficients, -79 and 1601, is -126479.

Considering quadratics of the form:

**, where**

*n*+^{2}*an*+*b***|**and

*a*|<1000**|**, where

*b*|<1000**|**is the modulus/absolute value of

*n*|*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.

#### The Catch

How to speed up the search from **| a| < 1000** and

**|**That is looping from -999 to 999 for a and b, totaling 1999 * 1999 = 3,996,001 iterations!

*b*| < 1000.#### The Light

By doing a little mathematical analysis, it is obvious that for n = 0, n^{2} + an + b = 0^{2} + 0 + b = b. Since this function must return a prime number, b must be prime. Thus, we can skip all non-prime values for b and keep track of the streak. It can be shown that this simple tweak lowers the number of iterations to 355,822. Use the prime checking method discussed in Problem 3 to check for prime number.

#### The Code

public class Problem27 { public static void main(String[] args) { int maxA = 0, maxB = 0 , maxN = 0; int count = 0; for(int a = -999; a < 1000; a++) { for(int b = -999; b < 1000; b++) { if(!isPrime(b)) continue; int n = 0; while(isPrime( n*n + a*n + b )) { n++; } if(n > maxN) { maxN = n; maxA = a; maxB = b; } } } System.out.println("a * b = " + (maxA * maxB)); } public static boolean isPrime(int n) { if( n < 2 ) return false; if(n % 2 == 0 && n != 2) return false; for(int i = 3; i * i < n; i++) { if(n % i == 0) return false; } return true; } }