1. Introduction. For a prime p, the well-known Wilson congruence says that (p − 1)! ≡ −1 modulo p. A prime p is called a Wilson prime if the congruence above holds modulo p 2 . We now quote from [R96, pp. 346, 350]: 'It is not known whether there are infinitely many Wilson primes. In this respect, Vandiver wrote: This question seems to be of such a character that if I should come to life any time after my death and some mathematician were to tell me it had been definitely settled, I think I would immediately drop dead again.' Ribenboim also mentions that search (by Crandall, Dilcher, Pomerance [CDP97]) up to 5 • 10 8 produced the only known Wilson primes, namely 5, 13, and 563, as discovered by Goldberg in 1953 (one of the first successful computer searches involving very large numbers). See [R96, Dic19] for other historical references.Many strong analogies [Gos96,Ro02,Tha04] between number fields and function fields over finite fields have been used to benefit the study of both. These analogies are even stronger in the base case Q, Z ↔ F (t), F [t], where F is a finite field. We study the concept of Wilson prime in this function field context, and in contrast to the Z case, we exhibit infinitely many of them, at least for many F . For example, ℘ = t 3 * 13 n − t 13 n − 1 are Wilson primes for F 3 [t].We also show strong connections between Wilson's and Fermat's quotients, and also between refined Wilson residues and discriminants. Moreover, we introduce analogs of Bell numbers in the F [t] setting.2. Wilson primes. Let us fix some basic notation. We use the standard conventions that empty sums are zero and that empty products are one.

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