On a congruence of Emma Lehmer related to Euler numbers

Volume 161 / 2013

John B. Cosgrave, Karl Dilcher Acta Arithmetica 161 (2013), 47-67 MSC: Primary 11A07; Secondary 11B68. DOI: 10.4064/aa161-1-4


A congruence of Emma Lehmer (1938) for Euler numbers $E_{p-3}$ modulo $p$ in terms of a certain sum of reciprocals of squares of integers was recently extended to prime power moduli by T. Cai et al. We generalize this further to arbitrary composite moduli $n$ and characterize those $n$ for which the sum in question vanishes modulo $n$ (or modulo $n/3$ when $3\,|\, n$). Primes for which $E_{p-3}\equiv 0\pmod{p}$ play an important role, and we present some numerical results.


  • John B. Cosgrave79 Rowanbyrn
    Blackrock, County Dublin, Ireland
  • Karl DilcherDepartment of Mathematics and Statistics
    Dalhousie University
    Halifax, NS, B3H 3J5, Canada

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