MathLabs

Problem 3

Let k,m,nk,m,n be natural numbers such that m+k+1m+k+1 is a prime greater than n+1n+1. Put cs=s(s+1)c_s=s(s+1). Prove that (cm+1−ck)(cm+2−ck)⋯(cm+n−ck)(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots(c_{m+n}-c_k) is divisible by c1c2⋯cnc_1c_2\cdots c_n.
Step 2 of 6: Rewrite the whole product
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P=∏i=1n(cm+i−ck)=(∏i=1n(m−k+i))(∏i=1n(m+k+i+1))P=\prod_{i=1}^n(c_{m+i}-c_k)=\left(\prod_{i=1}^n(m-k+i)\right)\left(\prod_{i=1}^n(m+k+i+1)\right)
Detailed analysis

Applying the factorization with a=m+ia=m+i and b=kb=k gives two blocks of nn consecutive integers.