Can one classify all solutions of$$\prod _{1\le i\le k_1}(m_1+i)=\prod _{1\le j\le k_2}(m_2+j)$$where $k_1,k_2>3$ and $m_1+k_1\le m_2$? Are there only finitely many solutions?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write the equation as a “factorial–ratio” identity: [ \prod_{i=1}^{k_1}(m_1+i)=\frac{(m_1+k_1)!}{m_1!},\qquad \prod_{j=1}^{k_2}(m_2+j)=\frac{(m_2+k_2)!}{m_2!}, ] so your equation is [ \frac{(m_1+k_1)!}{m_1!}=\frac{(m_2+k_2)!}{m_2!} \quad\Longleftrightarrow\quad (m_1+k_1)!,m_2!=(m_2+k_2)!,m_1!. ] The condition (m_1+k_1\…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.