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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 See [R96, Dic19] for other historical references.Many strong analogies [Gos96,Ro02,Tha04] between number fields and function fields over finite fields have been used to benefit the study of both.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.

Although there has been a number of experimental determinations of the thermal conductivity of air, its value even to two figures is still uncertain. It is true that the values hitherto obtained are of the same order as that deduced from the kinetic theory by Clausius, Maxwell, and others, but they differ among themselves to the extent of 20 per cent. There have been only two methods of experiment. Stefan, Kundt and Warburg, and Winkelmann observed the rate of cooling of a thermometer in an envelope of rarefied air, the outer surface of which was kept at a constant temperature. Schleier-macher and Schwarze measured the rate at which heat passed from an electrically heated wire stretched along the axis to the surface of a cylinder maintained at a lower constant temperature. Repetitions of these two methods have not cleared away the uncertainty as to what is the correct value of the thermal conductivity of air. Under these circumstances it might be interesting to give an account of another method for determining this quantity. It will be noticed that in this method, which was suggested by Prof. Poynting, the conductivity is measured at ordinary pressures, there being no need to take precautions against convection currents by the reduction of the pressure as is necessary in the other methods.

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

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