# Prime Factors of 1000 Factorial

## Example of Factorial

The prime decomposition of $1000!$ is given as:

 $\ds 1000!$ $=$ $\ds 2^{994} \times 3^{498} \times 5^{249} \times 7^{164} \times 11^{98} \times 13^{81} \times 17^{61} \times 19^{54} \times 23^{44} \times 29^{35} \times 31^{33} \times 37^{27}$ $\ds$  $\, \ds \times \,$ $\ds 41^{24} \times 43^{23} \times 47^{21} \times 53^{18} \times 59^{16} \times 61^{16} \times 67^{14} \times 71^{14} \times 73^{13} \times 79^{13} \times 83^{12} \times 89^{11}$ $\ds$  $\, \ds \times \,$ $\ds 97^{10} \times 101^9 \times 103^9 \times 107^9 \times 109^9 \times 113^8 \times 127^7 \times 131^7 \times 137^7 \times 139^7 \times 149^6 \times 151^6$ $\ds$  $\, \ds \times \,$ $\ds 157^6 \times 163^6 \times 167^5 \times 173^5 \times 179^5 \times 181^5 \times 191^5 \times 193^5 \times 197^5 \times 199^5 \times 211^4 \times 223^4$ $\ds$  $\, \ds \times \,$ $\ds 227^4 \times 229^4 \times 233^4 \times 239^4 \times 241^4 \times 251^3 \times 257^3 \times 263^3 \times 269^3 \times 271^3 \times 277^3 \times 281^3$ $\ds$  $\, \ds \times \,$ $\ds 283^3 \times 293^3 \times 307^3 \times 311^3 \times 313^3 \times 317^3 \times 331^3 \times 337^2 \times 347^2 \times 349^2 \times 353^2 \times 359^2$ $\ds$  $\, \ds \times \,$ $\ds 367^2 \times 373^2 \times 379^2 \times 383^2 \times 389^2 \times 397^2 \times 401^2 \times 409^2 \times 419^2 \times 421^2 \times 431^2 \times 433^2$ $\ds$  $\, \ds \times \,$ $\ds 439^2 \times 443^2 \times 449^2 \times 457^2 \times 461^2 \times 463^2 \times 467^2 \times 479^2 \times 487^2 \times 491^2 \times 499^2 \times 503$ $\ds$  $\, \ds \times \,$ $\ds 509 \times 521 \times 523 \times 541 \times 547 \times 557 \times 563 \times 569 \times 571 \times 577 \times 587 \times 593$ $\ds$  $\, \ds \times \,$ $\ds 599 \times 601 \times 607 \times 613 \times 617 \times 619 \times 631 \times 641 \times 643 \times 647 \times 653 \times 659$ $\ds$  $\, \ds \times \,$ $\ds 661 \times 673 \times 677 \times 683 \times 691 \times 701 \times 709 \times 719 \times 727 \times 733 \times 739 \times 743$ $\ds$  $\, \ds \times \,$ $\ds 751 \times 757 \times 761 \times 769 \times 773 \times 787 \times 797 \times 809 \times 811 \times 821 \times 823 \times 827$ $\ds$  $\, \ds \times \,$ $\ds 829 \times 839 \times 853 \times 857 \times 859 \times 863 \times 877 \times 881 \times 883 \times 887 \times 907 \times 911$ $\ds$  $\, \ds \times \,$ $\ds 919 \times 929 \times 937 \times 941 \times 947 \times 953 \times 967 \times 971 \times 977 \times 983 \times 991 \times 997$

## Proof

For each prime factor $p$ of $1000!$, let $a_p$ be the integer such that:

$p^{a_p} \divides 1000!$
$p^{a_p + 1} \nmid 1000!$

Taking the prime factors in turn:

### Multiplicity of $2$

The prime factor $2$ appears in $1000!$ to the power of $994$.

That is:

$2^{994} \divides 1000!$

but:

$2^{995} \nmid 1000!$

### Multiplicity of $3$

The prime factor $3$ appears in $1000!$ to the power of $498$.

That is:

$3^{498} \divides 1000!$

but:

$3^{499} \nmid 1000!$

### Multiplicity of $5$

The prime factor $5$ appears in $1000!$ to the power of $249$.

That is:

$5^{249} \divides 1000!$

but:

$5^{250} \nmid 1000!$

### Multiplicity of $7$

The prime factor $7$ appears in $1000!$ to the power of $164$.

That is:

$7^{164} \divides 1000!$

but:

$7^{165} \nmid 1000!$

### Multiplicity of $11$

The prime factor $11$ appears in $1000!$ to the power of $98$.

That is:

$11^{98} \divides 1000!$

but:

$11^{99} \nmid 1000!$

### Multiplicity of $13$

The prime factor $13$ appears in $1000!$ to the power of $81$.

That is:

$13^{81} \divides 1000!$

but:

$13^{82} \nmid 1000!$

And so on: for the remaining factors:

 $\ds a_{17}$ $=$ $\ds \sum_{k \mathop > 0} \floor {\frac {1000} {17^k} }$ De Polignac's Formula $\ds$ $=$ $\ds \floor {\frac {1000} {17} } + \floor {\frac {1000} {289} }$ $\ds$ $=$ $\ds 58 + 3$ $\ds$ $=$ $\ds 61$

 $\ds a_{19}$ $=$ $\ds \sum_{k \mathop > 0} \floor {\frac {1000} {19^k} }$ De Polignac's Formula $\ds$ $=$ $\ds \floor {\frac {1000} {19} } + \floor {\frac {1000} {361} }$ $\ds$ $=$ $\ds 52 + 2$ $\ds$ $=$ $\ds 54$

 $\ds a_{23}$ $=$ $\ds \sum_{k \mathop > 0} \floor {\frac {1000} {23^k} }$ De Polignac's Formula $\ds$ $=$ $\ds \floor {\frac {1000} {23} } + \floor {\frac {1000} {529} }$ $\ds$ $=$ $\ds 43 + 1$ $\ds$ $=$ $\ds 44$

 $\ds a_{29}$ $=$ $\ds \sum_{k \mathop > 0} \floor {\frac {1000} {29^k} }$ De Polignac's Formula $\ds$ $=$ $\ds \floor {\frac {1000} {29} } + \floor {\frac {1000} {841} }$ $\ds$ $=$ $\ds 34 + 1$ $\ds$ $=$ $\ds 35$

 $\ds a_{31}$ $=$ $\ds \sum_{k \mathop > 0} \floor {\frac {1000} {31^k} }$ De Polignac's Formula $\ds$ $=$ $\ds \floor {\frac {1000} {31} } + \floor {\frac {1000} {961} }$ $\ds$ $=$ $\ds 32 + 1$ $\ds$ $=$ $\ds 33$

 $\ds a_{37}$ $=$ $\ds \sum_{k \mathop > 0} \floor {\frac {1000} {37^k} }$ De Polignac's Formula $\ds$ $=$ $\ds \floor {\frac {1000} {37} }$ $\ds$ $=$ $\ds 27$

The pattern is clear: the remaining indices are calculated in the same way.

$\blacksquare$