On a congruence of Emma Lehmer related to Euler numbers (Q2855865)

The paper completes the 2013 trilogy of \textit{J. B. Cosgrave} and \textit{K. Dilcher} [J. Number Theory 133, No. 11, 3565--3577 (2013; Zbl 1295.11030)], [Fibonacci Q. 51, No. 2, 98--111 (2013; Zbl 1325.11005)] by extending to arbitrary composite moduli a congruence modulo prime powers due to \textit{T.-X. Cai} et al. [Acta Arith. 130, No. 3, 203--214 (2007; Zbl 1135.11002)] and derived from the sum of reciprocal squares involving Euler numbers NEWLINE\[NEWLINE \sum _{j=1}^{[p/4]} \frac{1}{j^2} \equiv (-1)^{\frac{(p-1)}{2}} 4E_{p-3} \pmod{p} NEWLINE\]NEWLINE originally found by \textit{E. Lehmer} [Ann. Math. (2) 39, 350--360 (1938; Zbl 0019.00505)].NEWLINENEWLINEThe extended sum of reciprocals vanishes modulo \(n\) (or modulo \(n/3\) when \(3\,|\, n\)) for characterized \(n\) and is applicable modulo \(n^2\); the authors remark the role of primes \(p \geq 5\) for which \(E_{p-3}\equiv 0\pmod{p}\) supplying new calculations up to 50 million and reporting further independent computations up to \(3 \cdot 10^9\).NEWLINENEWLINEThe authors also recall the Kummer congruence for Euler numbers described in [\textit{F. W. J. Olver} (ed.) et al., NIST handbook of mathematical functions. Cambridge: Cambridge University Press (2010; Zbl 1198.00002)] and its generalizations (explored, e.g., by \textit{P. T. Young} [J. Number Theory 78, No. 2, 204--227 (1999; Zbl 0939.11014)], by \textit{Z.-H. Sun} [Bull. Aust. Math. Soc. 82, No. 2, 221--231 (2010; Zbl 1203.11027)], by \textit{L. Carlitz} [Nagoya Math. J. 7, 35--43 (1954; Zbl 0056.26802)] and by \textit{L. Carlitz} and \textit{J. Levine} [Trans. Am. Math. Soc. 96, 23--37 (1960; Zbl 0099.02902)]) clarifying the importance, for their work, of the refinements provided by \textit{K.-W. Chen} [Fibonacci Q. 42, No. 2, 128-140 (2004; Zbl 1067.11008)] and by \textit{Y. He} and \textit{Q.-Y. Liao} [Fibonacci Q. 46--47, No. 3, 225--234 (2009; Zbl 1220.11029)].NEWLINENEWLINEBeyond Euler numbers and quotients and the Euler's generalization of Fermat's Little Theorem, the proof employs the Fermat quotients investigated by \textit{M. Lerch} [Math. Ann. 60, 471--490 (1905; JFM 36.0266.03)] and by \textit{Z.-H. Sun} [J. Number Theory 128, No. 2, 280--312 (2008; Zbl 1154.11010)], an identity given by \textit{T.-X. Cai} [Acta Arith. 103, No. 4, 313--320 (2002; Zbl 1008.11001)], an approach (alternative to the ``inclusion-exclusion'' argument) via the Möbius function developed by \textit{H.-Q. Cao} and \textit{H. Pan} [J. Number Theory 129, No. 8, 1813--1819 (2009; Zbl 1246.11004)], the Chinese Remainder Theorem and other properties illustrated by \textit{I. Niven} et al. [An introduction to the theory of numbers. 5th ed. New York etc.: Wiley (1991; Zbl 0742.11001)].