Let $p=4k+1,$$p$ is a prime numbe,and $ k\in\mathbb{Z}$. Prove the existence of positive integer $a_1,a_2,\ldots,a_k$ and $b_1,b_2,\ldots,b_k$. $$ p=\frac{(a^{2}_{1}+1)(a^{2}_{2}+1)\cdots(a^{2}_{k}+1)}{(b_{1}^{2}+1)(b_{2}^{2}+1)\cdots(b_{k}^{2}+1)}. $$ My friend trying to prove the following for a long time,and I cann't solve it,I hope you can help me. I would be extremely happy if somebody could help me with this!
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
|||||||||||||||||||||
|
|
Maybe something like this works. There's an integer $a_1$ such that $a_1^2+1=pm_1$ with $m_1\lt p$. There's an integer $b_1$ such that $b_1^2+1=m_1n_2$ with $n_2\lt m_1$. There's an integer $a_2$ with $a_2^2+1=n_2m_2$ with $m_2\lt n_2$. Etc. The sequence $p\gt m_1\gt n_2\gt m_2\dots$ must terminate after at most $k$ steps (since each $m_i$ and $n_i$ is 1 mod 4). |
|||
|
|
