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 4 of 6: Introduce the prime
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p=m+k+1,B=∏i=1n(p+i),(p+nn+1)=p B(n+1)!p=m+k+1,\quad B=\prod_{i=1}^n(p+i),\quad \binom{p+n}{n+1}=p\,\frac{B}{(n+1)!}
Detailed analysis

Set p=m+k+1p=m+k+1. The second block is B=∏i=1n(p+i)B=\prod_{i=1}^n(p+i), and the displayed identity follows by expanding the binomial coefficient.