ACTA ARITHMETICA * (201*) On a congruence of Emma Lehmer related to Euler numbers by John B. Cosgrave (Blackrock) and Karl Dilcher (Halifax) 1. Introduction. Congruences for sums of reciprocals modulo a prime or prime power have been of considerable interest throughout the 20th cen- tury, mainly because of their close connection to the first case of Fermat’s Last Theorem; see, e.g., [17, pp. 155 ff.] or [11]. Even though this motivation is now only of historical interest, such congruences have continued to attract attention, and have recently been extended to composite moduli; see [1], [2] or [3]. A brief historical overview is given in [7], where some of the earlier results, relating sums of reciprocals with Fermat and Euler quotients, have been further extended. All these papers are based on methods and results of Emma Lehmer, whose 1938 paper [11] remains the most important and influential paper on this topic, although it built on earlier work of Glaisher, Lerch and oth- ers. One of the more remarkable congruences in Lehmer’s paper [11] is the following one for sums of reciprocals of squares: (1.1) ⌊p/4⌋  j =1 1 j 2 ≡ (−1) (p−1)/2 4E p−3 (mod p), valid for all primes p ≥ 5, where E n is the nth Euler number, which can be defined by the exponential generating function (1.2) 2 e t + e −t = ∞  n=0 E n n! t n (|t| <π). Recently Cai, Fu and Zhou [2] extended (1.1) to prime powers by proving the following congruence for odd primes p and integers α ≥ 1: (1.3) ⌊p α /4⌋  j =1 p∤j 1 j 2 ≡ (−1) (p α −1)/2 4E ϕ(p α )−2  (mod p α ) when p ≥ 5, (mod 3 α−1 ) when p =3. 2010 Mathematics Subject Classification : Primary 11A07; Secondary 11B68. Key words and phrases : sums of reciprocals, congruences, Euler numbers. [1]