From 0fa5d5c81b99f874d95f4fcf893951fad2616846 Mon Sep 17 00:00:00 2001
From: moophis <moophis@gmail.com>
Date: Sat, 5 Jan 2013 10:07:40 +0800
Subject: [PATCH] Problem 27: Quadratic primes

---
 Problem 27.cpp | 155 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 155 insertions(+)
 create mode 100644 Problem 27.cpp

diff --git a/Problem 27.cpp b/Problem 27.cpp
new file mode 100644
index 0000000..7e6ff1b
--- /dev/null
+++ b/Problem 27.cpp	
@@ -0,0 +1,155 @@
+/*Euler published 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.
+
+ Using computers, the incredible formula  n^2 - 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.
+
+ Considering quadratics of the form:
+
+ n^2 + 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.
+ */
+#include <iostream>
+#include <cmath>
+#include <vector>
+#include <cassert>
+
+using namespace std;
+
+vector<int> prime_tab;
+
+bool is_prime(int n)
+{
+	n = abs(n);
+	if (n == 0) return false;
+	if (n == 1) return false;
+	else if (n < 4) return true;
+	else if (n % 2 == 0) return false;
+	else if (n < 9) return true;
+	else if (n % 3 == 0) return false;
+	else
+	{
+	    int r = (int) sqrt(n);
+		for (int i = 5; i <= r; i = i + 6)
+		{
+			if (n % i == 0) return false;
+			if (n % (i + 2) == 0) return false;
+		}
+	}
+	return true;
+}
+
+
+void make_primetab(int range)
+{
+	assert(range > 0);
+
+	for (int i = 0; i < range; i++) {
+		if ( is_prime(i) )
+			prime_tab.push_back(i);
+	}
+}
+
+int do_coefficient_mul()
+{
+	int a, b, cur_a, cur_b;
+	int cnt = 2;		// by default, P(0), P(1) are prime
+	int maxval = 0;
+	int ans;
+
+	make_primetab(2000);
+	/*
+	 * Role 1: b = P(0) must be a prime number;
+	 * Role 2: |1 + a + b| = |p(1)| must be a prime number.
+	 */
+	for (int i = 0; i < prime_tab.size(); i++) {
+		if ( prime_tab[i] >= 1000 )
+			break;
+			
+		// when b > 0, P(1) > 0
+		b = prime_tab[i];
+		for (int j = 0; j < prime_tab.size(); j++) {
+			a = prime_tab[j] - b - 1;
+			cnt = 2;
+			while ( is_prime(cnt * cnt + a * cnt + b) )
+				cnt++;
+			if (cnt > maxval) {
+				maxval = cnt;
+				cur_a = a;
+				cur_b = b;
+				ans = a * b;
+			}
+		}
+
+		// when b > 0, P(1) < 0
+		for (int j = 0; j < p_num; j++) {
+			a = -prime_tab[j] - b - 1;
+			cnt = 2;
+			if (abs(a) >= 1000)
+				break;
+			while ( is_prime(cnt * cnt + a * cnt + b) )
+				cnt++;
+			if (cnt > maxval) {
+				maxval = cnt;
+				cur_a = a;
+				cur_b = b;
+				ans = a * b;
+			}
+		}
+
+		// when b < 0, P(1) > 0
+		b = -prime_tab[i];
+		for (int j = 0; j < p_num; j++) {
+			a = prime_tab[j] - b - 1;
+			cnt = 2;
+			if (a >= 1000)
+				break;
+			while ( is_prime(cnt * cnt + a * cnt + b) )
+				cnt++;
+			if (cnt > maxval) {
+				maxval = cnt;
+				cur_a = a;
+				cur_b = b;
+				ans = a * b;
+			}
+		}
+
+		// when b < 0, P(1) < 0
+		for (int j = 0; j < p_num; j++) {
+			a = -prime_tab[j] - b - 1;
+			cnt = 2;
+			while ( is_prime(cnt * cnt + a * cnt + b) )
+				cnt++;
+			if (cnt > maxval) {
+				maxval = cnt;
+				cur_a = a;
+				cur_b = b;
+				ans = a * b;
+			}
+		}
+	}
+
+	cout << "a = " << cur_a << " b = " << cur_b
+		 << " maxval = " << maxval << endl;
+	return ans;
+}
+
+int main()
+{
+	cout << do_coefficient_mul() << endl;
+	return 0;
+}