Prime Factors of 52 Factorial
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Example of Factorial
The prime decomposition of $52!$ is given as:
- $52! = 2^{49} \times 3^{23} \times 5^{12} \times 7^8 \times 11^4 \times 13^4 \times 17^3 \times 19^2 \times 23^2 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47$
Proof
For each prime factor $p$ of $52!$, let $a_p$ be the integer such that:
- $p^{a_p} \divides 52!$
- $p^{a_p + 1} \nmid 52!$
Taking the prime factors in turn:
\(\ds a_2\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {2^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} 2} + \floor {\frac {52} 4} + \floor {\frac {52} 8 } + \floor {\frac {52} {16} } + \floor {\frac {52} {32} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 26 + 13 + 6 + 3 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 49\) |
\(\ds a_3\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {3^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} 3} + \floor {\frac {52} 9} + \floor {\frac {52} {27} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 17 + 5 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 23\) |
\(\ds a_5\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {5^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} 5} + \floor {\frac {52} {25} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12\) |
\(\ds a_7\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {7^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} 7} + \floor {\frac {52} {49} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8\) |
\(\ds a_{11}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {11^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} {11} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4\) |
\(\ds a_{13}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {13^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} {13} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4\) |
\(\ds a_{17}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {17^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} {17} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
\(\ds a_{19}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {19^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} {19} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2\) |
\(\ds a_{23}\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {52} {23^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {52} {23} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2\) |
Similarly:
- $a_{29} = 1$
- $a_{31} = 1$
- $a_{37} = 1$
- $a_{41} = 1$
- $a_{43} = 1$
- $a_{47} = 1$
Hence the result.
$\blacksquare$