Prime Decomposition of 10th Fermat Number/Historical Note
Historical Note on Prime Decomposition of 10th Fermat Number
$F_{10}$ was proved composite in $1952$ by Raphael Mitchel Robinson using Pépin's Test on the SWAC, but at that time the factors had yet to be determined.
John Lewis Selfridge discovered the factor $45 \, 592 \, 577$ in $1953$, also using the SWAC.
At the same time he discovered the factor $825 \, 753 \, 601$ of $F_{16}$.
John David Brillhart discovered the factor $6 \, 487 \, 031 \, 809$ in $1962$ on an IBM 704.
Brillhart later found that the cofactor was a $291$-digit composite number.
The factors of this $291$-digit composite were finally found by Richard Peirce Brent in $1995$.
As he explained in his $1999$ article, the reason why the factors of $F_{11}$ were found so much earlier is that the second largest factor of $F_{11}$ had a mere $22$ digits, as opposed to the $40$ digits of that of $F_{10}$.
It may be noted that the author of this page performed this exercise in $2020$ using a factorisation tool freely available online, running on a machine of modest specifications.
It took just under $7$ hours in total.
Sources
- 1954: R.M. Robinson: Mersenne and Fermat numbers (Proc. Amer. Math. Soc. Vol. 5: pp. 842 – 846) www.jstor.org/stable/2031878
- 1953: J.L. Selfridge: Note $156$ -- Factors of Fermat numbers (MTAC Vol. 7: pp. 274 – 275) www.jstor.org/stable/2002843
- 1963: John Brillhart: Some miscellaneous factorizations (Math. Comp. Vol. 17: pp. 447 – 450) www.jstor.org/stable/2004009
- Jan. 1999: Richard P. Brent: Factorization of the Tenth Fermat Number (Math. Comp. Vol. 68, no. 235: pp. 429 – 451) www.jstor.org/stable/2585124