Factorial/Examples/1000
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Example of Factorial
The factorial of $1\,000$ starts:
- $402,387,260,077 \ldots$
and has $2568$ digits, of which the last $249$ are $0$.
Prime Factors of $1000!$
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
From Stirling's Formula:
- $n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
whence an approximate value for $1 \, 000!$ can be calculated.
Let $d$ be the number of digits in $1\,000!$
From Number of Digits in Factorial:
- $d = 1 + \floor {\paren {n + \dfrac 1 2} \log_{10} n - 0.43429 \ 4481 \, n + 0.39908 \ 9934}$
from which the result can be calculated by setting $n = 1000$.
From Prime Factors of $1000!$:
- the multiplicity of $5$ in $1000!$ is $249$
- the multiplicity of $2$ in $1000!$ is $994$.
Therefore there is a factor of $10^{249}$ in $1000!$, but not $10^{250}$.
Hence there are $249$ instances of $0$ at the end of $1000!$.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $4$