"

 Fermat aware of Euler's result 

I've added reference Křížek, Luca, Somer 2001, p. 38, Remark 4.15 to the claim that Fermat probably knew this result. However, the claim that it is ""widely believed"" might be too strong. The mentioned remarks says the following:
Remark 4.15. As mentioned in the literature (cf., e.g., [Lenstra, Lenstra, Manasse, Pollard]), Fermat probably knew the previous theorem, although its proof was given later by Euler in 1747 (see [Burton, p.219]). However, it is then unclear why Fermat did not apply this theorem to the factorization of F5. It was sufficient to divide F5 only by primes of the form 64k + 1, and for k = 10 Fermat could have convinced himself that his hypothesis on the primality of Fm did not hold. Concerning the factor 2142.27 + 1 = 274177 of F6 (see Figure 1.2), we note that there are 370 primes of the form k27 + 1 for k < 2142. 
  "