# Prime Factors of 52 Factorial

## 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.

$\blacksquare$