Here is the well-known factorisation of F5, the 5th Fermat number, and the first one to be composite:
F5 = 2^32 + 1 = 641 * 6700417
In my paper 18, I gave a mnemonic for this factorisation:
"Euler's left a broken theorem . . what a tragedy!
Note that five decimal digits each occur twice, and the other five not at all. I recently asked what was the probability that ten random decimal digits will have that property. The answer is:
9*7*5*3*9*8*7*6/10^9 = 0.00285768
For comparison, the probability that ten random digits will all be different is:
10!/10^10 = 0.00036288
In the table below we give the probabilities that if we choose 2n random decimal digits:
i. we will get n digits each repeated twice,
ii. we will get not more than n different digits:
2n. Prob. 1 Prob. 2
2. 0.1 0.1
4. 0.027 0.064
6. 0.0108 0.0676
8. 0.005292 0.0928
10. 0.00285768 0.14646088
(Column 1 is easy to calculate. Column 2 was found by counting. I thought that the last entry in column 2 was surprisingly large, so I checked it a different way)
(Last modified 20th Feb. 2025)