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Suppose $n\geq 5$ is an odd positive integer. Prove that $${n \choose 1}-5{n \choose 2}+5^2{n \choose 3}-...+5^{n-1}{n\choose n}$$ is not a prime number.

I tried expanding each to see if anything jumped out at me to prove this. I am struggling. Any help or hints would be much appreciated.

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    $\begingroup$ Hint: Expand $(5-1)^n$ using the binomial theorem. $\endgroup$
    – Lucian
    Mar 10, 2015 at 1:43
  • $\begingroup$ Thank you! This helped, it finally clicked. :) $\endgroup$
    – Dawn AM
    Mar 10, 2015 at 14:18

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Let \begin{align} \phi_{n} = {n \choose 1}-5{n \choose 2}+5^2{n \choose 3}-...+5^{n-1}{n\choose n} \end{align} then \begin{align} -5 \phi_{n} &= \sum_{r=1}^{n} \binom{n}{r} (-5)^{r} = -1 + \sum_{r=0}^{n} \binom{n}{r} (-5)^{r} \\ &= (1-5)^{n} -1 \end{align} or \begin{align} \phi_{n} = \frac{ 1 -(-4)^{n} }{ 5 }. \end{align} For the case of $n=5$ the result is $\phi_{5} = 205 = 5 \cdot 41$ which is the product of two primes. Further results show that $\phi_{n \geq 5}$ is not prime.

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