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.

$\blacksquare$