Let
![n\ge2](/media/m/0/4/2/04281c4af6de10a93a8cbff53cb03a56.png)
be an integer. Prove that if
![k^2+k+n](/media/m/2/c/e/2ce0aa94d5c42b768a531c0893559d9d.png)
is prime for all integers
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
such that
![0\le k\le\sqrt{n\over3}](/media/m/a/1/a/a1a7a2b4c5db9ae4e583cb09f54232f9.png)
, then
![k^2+k+n](/media/m/2/c/e/2ce0aa94d5c42b768a531c0893559d9d.png)
is prime for all integers
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
such that
![0\le k\le n-2](/media/m/5/9/5/595a471080863ef2745361735866714e.png)
.(IMO Problem 6)
Original Formulation
Let
![f(x) = x^2 + x + p](/media/m/b/c/2/bc2828a358c677f6b1bf8b7ab5560c63.png)
,
![p \in \mathbb N.](/media/m/b/6/e/b6eee6de7673999a0b2971b171e2b154.png)
Prove that if the numbers
![f(0), f(1), \cdots , f(\sqrt{p\over 3} )](/media/m/f/2/4/f24df54334a73db1b9cb1987caedbede.png)
are primes, then all the numbers
![f(0), f(1), \cdots , f(p - 2)](/media/m/f/a/0/fa01a6b5df4365e616c924dcb73350a3.png)
are primes.
Proposed by Soviet Union.
%V0
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.(IMO Problem 6)
Original Formulation
Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.
Proposed by Soviet Union.