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A congruence of Emma Lehmer (1938) for Euler numbers 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 play an important role, and we present some numerical results.
John B. Cosgrave, and Karl Dilcher. "On a congruence of Emma Lehmer related to Euler numbers." Acta Arithmetica 161.1 (2013): 47-67. <http://eudml.org/doc/286382>.
@article{JohnB2013, abstract = {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 $E_\{p-3\} ≡ 0 (mod p)$ play an important role, and we present some numerical results.}, author = {John B. Cosgrave, Karl Dilcher}, journal = {Acta Arithmetica}, keywords = {congruences; Euler numbers; sums of reciprocals}, language = {eng}, number = {1}, pages = {47-67}, title = {On a congruence of Emma Lehmer related to Euler numbers}, url = {http://eudml.org/doc/286382}, volume = {161}, year = {2013}, }
TY - JOUR AU - John B. Cosgrave AU - Karl Dilcher TI - On a congruence of Emma Lehmer related to Euler numbers JO - Acta Arithmetica PY - 2013 VL - 161 IS - 1 SP - 47 EP - 67 AB - 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 $E_{p-3} ≡ 0 (mod p)$ play an important role, and we present some numerical results. LA - eng KW - congruences; Euler numbers; sums of reciprocals UR - http://eudml.org/doc/286382 ER -