It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
\[9 = 7 + 2\times 1^2\] \[15 = 7 + 2\times 2^2\] \[21 = 3 + 2\times 3^2\] \[25 = 7 + 2\times 3^2\] \[27 = 19 + 2\times 2^2\] \[33 = 31 + 2\times 1^2\]
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
哥德巴赫的另一个猜想
克里斯蒂安·哥德巴赫曾经猜想，每个奇合数都可以写成一个素数和一个平方的两倍之和。
\[9 = 7 + 2\times 1^2\] \[15 = 7 + 2\times 2^2\] \[21 = 3 + 2\times 3^2\] \[25 = 7 + 2\times 3^2\] \[27 = 19 + 2\times 2^2\] \[33 = 31 + 2\times 1^2\]
最终这个猜想被推翻了。
不能写成一个素数和一个平方的两倍之和的最小奇合数是多少？
没什么好说的，直接求就是了：
(defun eu7-isprime (n)
(cond
((<= n 1) nil)
((< n 4) t)
((zerop (% n 2)) nil)
((< n 9) t)
((zerop (% n 3)) nil)
(t (let ((bound (floor (sqrt n))))
(named-let f ((i 5))
(cond
((> i bound) t)
((zerop (% n i)) nil)
((zerop (% n (+ i 2))) nil)
(t (f (+ i 6)))))))))
(cl-block 'fin
(cl-loop for i from 35 by 2
if (not (eu7-isprime i)) do
(cl-loop for n from 1
if (> (* n n 2) i)
do (cl-return-from 'fin i) end
if (eu7-isprime (- i (* n n 2)))
return t)))
=> 5777