Prime Factors of 1000 Factorial
== Example of Factorial ==  The prime decomposition of $1000!$ is given as: {{begin-eqn}} {{eqn | l = 1000!       | r = 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} }} {{eqn | o =        | ro= \times       | r = 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}       | c =  }} {{eqn | o =        | ro= \times       | r = 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 }} {{eqn | o =        | ro= \times       | r = 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 }} {{eqn | o =        | ro= \times       | r = 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 }} {{eqn | o =        | ro= \times       | r = 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 }} {{eqn | o =        | ro= \times       | r = 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 }} {{eqn | o =        | ro= \times       | r = 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 }} {{eqn | o =        | ro= \times       | r = 509 \times 521 \times 523 \times 541 \times 547 \times 557 \times 563 \times 569 \times 571 \times 577 \times 587 \times 593 }} {{eqn | o =        | ro= \times       | r = 599 \times 601 \times 607 \times 613 \times 617 \times 619 \times 631 \times 641 \times 643 \times 647 \times 653 \times 659 }} {{eqn | o =        | ro= \times       | r = 661 \times 673 \times 677 \times 683 \times 691 \times 701 \times 709 \times 719 \times 727 \times 733 \times 739 \times 743 }} {{eqn | o =        | ro= \times       | r = 751 \times 757 \times 761 \times 769 \times 773 \times 787 \times 797 \times 809 \times 811 \times 821 \times 823 \times 827 }} {{eqn | o =        | ro= \times       | r = 829 \times 839 \times 853 \times 857 \times 859 \times 863 \times 877 \times 881 \times 883 \times 887 \times 907 \times 911 }} {{eqn | o =        | ro= \times       | r = 919 \times 929 \times 937 \times 941 \times 947 \times 953 \times 967 \times 971 \times 977 \times 983 \times 991 \times 997       | c =  }} {{end-eqn}}  == 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 $2$ === {{:De Polignac's Formula/Examples/2 in 1000}} === Multiplicity of $3$ === {{:De Polignac's Formula/Examples/3 in 1000}} === Multiplicity of $5$ === {{:De Polignac's Formula/Examples/5 in 1000}} === Multiplicity of $7$ === {{:De Polignac's Formula/Examples/7 in 1000}} === Multiplicity of $11$ === {{:De Polignac's Formula/Examples/11 in 1000}} === Multiplicity of $13$ === {{:De Polignac's Formula/Examples/13 in 1000}} And so on: for the remaining factors: {{begin-eqn}} {{eqn | l = a_{17}       | r = \sum_{k \mathop > 0} \floor {\frac {1000} {17^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {1000} {17} } + \floor {\frac {1000} {289} }       | c =  }} {{eqn | r = 58 + 3       | c =  }} {{eqn | r = 61       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{19}       | r = \sum_{k \mathop > 0} \floor {\frac {1000} {19^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {1000} {19} } + \floor {\frac {1000} {361} }       | c =  }} {{eqn | r = 52 + 2       | c =  }} {{eqn | r = 54       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{23}       | r = \sum_{k \mathop > 0} \floor {\frac {1000} {23^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {1000} {23} } + \floor {\frac {1000} {529} }       | c =  }} {{eqn | r = 43 + 1       | c =  }} {{eqn | r = 44       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{29}       | r = \sum_{k \mathop > 0} \floor {\frac {1000} {29^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {1000} {29} } + \floor {\frac {1000} {841} }       | c =  }} {{eqn | r = 34 + 1       | c =  }} {{eqn | r = 35       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{31}       | r = \sum_{k \mathop > 0} \floor {\frac {1000} {31^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {1000} {31} } + \floor {\frac {1000} {961} }       | c =  }} {{eqn | r = 32 + 1       | c =  }} {{eqn | r = 33       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a_{37}       | r = \sum_{k \mathop > 0} \floor {\frac {1000} {37^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {1000} {37} }       | c =  }} {{eqn | r = 27       | c =  }} {{end-eqn}} The pattern is clear: the remaining indices are calculated in the same way.