Here you can see all the Fermat number divisors on a logarithmic diagram. As you can see they are distributed almost uniformly. It apears that there are another 200 divisors waiting to be discovered inside the bounds of the above diagram.
Diagrams are available in MATHCAD format, click here.
This dependency can be expressed by the following formula. The number of Fermat divisors can be approximated by
where s is the correction factor. The behaviour of this dependency is a little complex.
| k | Known quantity | 0.8 ln(n).ln(k) | Divergence
|
|---|---|---|---|
| Status is completely known | |||
| 10^3 | 35 | 35 | 0 |
| 10^4 | 49 | 46 | +3 |
| 10^5 | 60 | 57 | +3 |
| 10^6 | 71 | 69 | +2 |
| 10^7 | 78 | 80 | -2 |
| 10^8 | 93 | 91 | +2 |
| 10^9 | 101 | 103 | -2 |
| 10^10 | 110 | 114 | -4 |
| Status is NOT completely known | |||
| 10^11 | 116 | 125 | -9 |
| 10^12 | 122 | 137 | -15 |
These results was collected on different hardware and softwares before, thus some mistakes may be present. There is a small probability, that for the calculated range not all numbers are found out. But the formula well describes known numbers. Increasing k in 10 times for all n<500, it is possible to find approximately 10 divisors! If we have increased k in 10 times for one number, then probability to find a divisor is 2%.
Guess. To find a Fermat divisor FOR ALMOST ALL n<500, we should increase k>10100.