Editing Fermat number (section)

===Heuristic arguments===
Heuristics suggest that ''F''<sub>4</sub> is the last Fermat prime.

The [[prime number theorem]] implies that a random integer in a suitable interval around ''N'' is prime with probability 1{{space|hair}}/{{space|hair}}ln ''N''. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that ''F''<sub>5</sub>, ..., ''F''<sub>32</sub> are composite, then the expected number of Fermat primes beyond ''F''<sub>4</sub> (or equivalently, beyond ''F''<sub>32</sub>) should be 
:<math> \sum_{n \ge 33} \frac{1}{\ln F_{n}} < \frac{1}{\ln 2} \sum_{n \ge 33} \frac{1}{\log_2(2^{2^n})} = \frac{1}{\ln 2} 2^{-32} < 3.36 \times 10^{-10}.</math>
One may interpret this number as an upper bound for the probability that a Fermat prime beyond ''F''<sub>4</sub> exists.

This argument is not a rigorous proof. For one thing, it assumes that Fermat numbers behave "randomly", but the factors of Fermat numbers have special properties. Boklan and [[John H. Conway|Conway]] published a more precise analysis suggesting that the probability that there is another Fermat prime is less than one in a billion.<ref>{{Cite journal |last1=Boklan |first1=Kent D. |last2=Conway |first2=John H. |date=2017 |title=Expect at most one billionth of a new Fermat Prime! |journal=The Mathematical Intelligencer |volume=39 |issue=1 |pages=3–5 |arxiv=1605.01371 |doi=10.1007/s00283-016-9644-3|s2cid=119165671 }}</ref>

Anders Bjorn and [[Hans Riesel]] estimated the number of square factors of Fermat numbers from ''F''<sub>5</sub> onward as
:<math>
\sum_{n \ge 5} \sum_{k \ge 1} \frac{1}{k (k 2^n + 1) \ln(k 2^n)} < \frac{\pi^2}{6 \ln 2} \sum_{n \ge 5} \frac{1}{n 2^n} \approx 0.02576;
</math>
in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of <math>a^{2^n} + b^{2^n}</math> are very rare for large ''n''.<ref name="bjorn">{{cite journal |last1=Björn |first1=Anders |last2=Riesel |first2=Hans |title=Factors of generalized Fermat numbers |journal=Mathematics of Computation |date=1998 |volume=67 |issue=221 |pages=441–446 |doi=10.1090/S0025-5718-98-00891-6 |url=https://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/ |language=en |issn=0025-5718|doi-access=free }}</ref>