# 404d59f863aa

`Problem 27`
author Steve Losh Wed, 22 Feb 2017 17:26:37 +0000 d84ef3399dd1 81710e614e21 (none) src/euler.lisp

## Changes

```--- a/src/euler.lisp	Wed Feb 22 17:13:41 2017 +0000
+++ b/src/euler.lisp	Wed Feb 22 17:26:37 2017 +0000
@@ -738,6 +738,38 @@
(for i :in (set-difference (primes-below 1000) '(2 5)))
(finding i :maximizing (multiplicative-order 10 i))))

+(defun problem-27 ()
+  ;; 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| ≤ 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.
+  (flet ((primes-produced (a b)
+           (iterate (for n :from 0)
+                    (while (primep (+ (square n) (* a n) b)))
+                    (counting t))))
+    (iterate (for-nested ((a :from -999 :to 999)
+                          (b :from -1000 :to 1000)))
+             (finding (* a b) :maximizing (primes-produced a b)))))
+

;;;; Tests --------------------------------------------------------------------
(def-suite :euler)
@@ -768,6 +800,7 @@
(test p23 (is (= 4179871 (problem-23))))
(test p24 (is (= 2783915460 (problem-24))))
(test p25 (is (= 4782 (problem-25))))
+(test p26 (is (= 983 (problem-26))))

;; (run! :euler)```