Euler discovered the remarkable quadratic formula:
n2 + n + 41
It turns out that the formula will produce 40 primes for the consecutive integer values 0 ≤ n ≤ 39. However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 412 + 41 + 41 is clearly divisible by 41.
The incredible formula n2 - 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:
n2 + an + b, where |a| < 1000 and |b| ≤ 1000where |n| is the modulus/absolute value of n
e.g. |11| = 11 and |-4| = 4Find 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.
using Common.Math; using System; using System.Diagnostics; using System.Linq; namespace ProjectEuler_27 { internal class Program { private static int[] primeCache = Eratosthenes.GetPrimes(1000).ToArray(); private static bool IsPrime(int num) { if (num > primeCache.Last()) { primeCache = Eratosthenes.GetPrimes(primeCache.Last() * 2); } // array.contains checks every value, these are in order from smallest to largest // so creating our own contains is more efficient. int i = 0; while (i < primeCache.Length && primeCache[i] < num) { i++; } return primeCache[i] == num; } private static void Main(string[] args) { Stopwatch sw = Stopwatch.StartNew(); ResultHelper result = new ResultHelper(); // 'b' must be prime because when n=0 we get 0^2 + a(0) + b which is just 'b' // since 0 is inclusive and all results must be prime, we can narrow down 'b' to just prime numbers int[] bPossibles = primeCache; // we can also narrow down 'a' possiblities because all primes except for 2 are odd // so if n=1 we get 1^2 + a(1) + b = 1 + a + b // since an odd added to an odd is always even, we know that for all cases except 2 'a' must be odd // in the case of 2 'a' must be even for (int a = -999; a < 1000; a += 2) { for (int bIdx = 0; bIdx < bPossibles.Length; bIdx++) { // handle the case of b=2 int workingA = bPossibles[bIdx] == 2 ? a - 1 : a; int n = 0; while (IsPrime(Math.Abs(n * n + workingA * n + bPossibles[bIdx]))) { n++; } if (result.Count < n) { result = new ResultHelper { A = a, B = bPossibles[bIdx], Count = n }; } } } sw.Stop(); Console.WriteLine($"{result.Count} primes found in {sw.ElapsedMilliseconds}ms when a={result.A} and b={result.B} with a product of {result.A * result.B}."); } private struct ResultHelper { public int A; public int B; public int Count; } } }
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