Prime Factors of 39 Factorial
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Example of Factorial
The prime decomposition of $39!$ is given as:
- $39! = 2^{35} \times 3^{18} \times 5^8 \times 7^5 \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29 \times 31 \times 37$
Proof
For each prime factor $p$ of $39!$, let $a_p$ be the integer such that:
- $p^{a_p} \divides 39!$
- $p^{a_p + 1} \nmid 39!$
Taking the prime factors in turn:
\(\ds a_2\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {2^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} 2} + \floor {\frac {39} 4} + \floor {\frac {39} 8 } + \floor {\frac {39} {16} } + \floor {\frac {39} {32} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 19 + 9 + 4 + 2 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 35\) |
\(\ds a_3\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {3^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} 3} + \floor {\frac {39} 9} + \floor {\frac {39} {27} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 + 4 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 18\) |
\(\ds a_5\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {5^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} 5} + \floor {\frac {39} {25} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8\) |
\(\ds a_7\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {7^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5\) |
\(\ds a_{11}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {11^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} {11} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
\(\ds a_{13}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {13^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} {13} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
\(\ds a_{17}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {17^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} {17} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2\) |
\(\ds a_{19}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {39} {19^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {39} {19} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2\) |
Similarly:
- $a_{23} = 1$
- $a_{29} = 1$
- $a_{31} = 1$
- $a_{37} = 1$
Hence the result.
$\blacksquare$