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Generalisation of a congruence of Gauss. (English) Zbl 0651.10004

For a prime \(p\equiv 1(4)\), \(p=a^2+b^2\), \(a\equiv 1(4)\) unique, Gauss proved that \(\binom{(p-1)/2}{(p-1)/4}\equiv 2a\pmod p\). An improvement of this congruence is \(\binom{(p-1)/2}{(p-1)/4}\equiv (-4)^{(p-1)/4}(a+bi)\pmod{p^2}\), where \(a+bi\in\mathbb Z_p\) and \(| a+bi|_p=1\). This was proved by S. Chowla, B. Dwork and R. Evans [J. Number Theory 24, 188–196 (1986; Zbl 0596.10003)]. Here the author generalizes this congruence to \[ \binom{(mp^{r-1})/2}{(mp^{r-1})/4}\equiv \binom{(mp^{r-1}-1)/2}{(mp^{r-1}-1)/4}(-4)^{\{mp^{r-1}\cdot (p-1)/4\}}\cdot (a+bi)\pmod {p^{2r}} \] where \(m\equiv 1(4)\). This congruence and many others are proved using Gauss and Jacobi sums and the theory of the \(p\)-adic \(\Gamma\)-function. The author remarks that these congruences are very easy to prove modulo \(p^r\) by the use of the theory of elliptic curves or the theory of formal group laws.

MSC:

11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)

Citations:

Zbl 0596.10003
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References:

[1] Atkin, A. O.L.; Swinnerton-Dyer, H. P.F., Modular forms on noncongruence subgroups, (Proc. Sympos. Pure Math., XIX (1971)), 1-25 · Zbl 0235.10015
[2] Berndt, B. C.; Evans, R. J., Sums of Gauss, Jacobi and Jacobsthal, J. Number Theory, 11, 349-398 (1979) · Zbl 0412.10027
[3] Beukers, F., Arithmetic properties of Picard-Fuchs equations, (Sem. Théorie Nombres Paris (1982-1983)), 33-38 · Zbl 0549.10003
[4] Chowla, S.; Dwork, B.; Evans, R. J., On the mod \(p^2\) determination of \((((p−1)2)((p−1)4))\), J. Number Theory, 24, 188-196 (1986) · Zbl 0596.10003
[5] Diamond, J., The \(p\)-adic log gamma function and \(p\)-adic Euler constants, Trans. Amer. Math. Soc., 233, 321-337 (1977) · Zbl 0382.12008
[6] C. Gauss; C. Gauss
[7] Gross, B.; Koblitz, N., Gauss-sums and the \(p\)-adic Γ-functions, Ann. Math., 109, 569-581 (1979) · Zbl 0406.12010
[8] Ireland, K.; Rosen, M., (A Classical Introduction to Modern Number Theory (1982), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0482.10001
[9] Iwasa, K., Lectures on \(p\)-adic \(L\)-functions, (Annals of Math. Studies (1972), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ)
[10] Koblitz, N., \(p\)-adic Numbers, \(p\)-adic Analysis, and the Zeta-Functions (1977), Springer-Verlag: Springer-Verlag New York/Heidelberg/Berlin · Zbl 0364.12015
[11] Koblitz, N., \(p\)-adic Analysis: A Short Course on Recent Work, (London Math. Soc. Lecture Notes 46 (1980), Cambridge Univ. Press: Cambridge Univ. Press London/New York) · Zbl 0439.12011
[12] Lidl, R.; Niederreiter, H., Finite Fields, (Encyclopedia of Math. and Its Applications (1983), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0866.11069
[13] Morita, Y., A \(p\)-adic analogue of the Γ-function, J. Fac. Sci. Univ. Tokyo, 22, 255-266 (1975) · Zbl 0308.12003
[14] Schikhof, W. H., (Ultrametric Calculus (1984), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0553.26006
[15] Coster, M. J., Supercongruences (1988), Leiden · Zbl 0715.11071
[16] L. van Hamme\(p\); L. van Hamme\(p\)
[17] Hardy, G. H.; Wright, E. M., (An Introduction to the Theory of Numbers (1938), Oxford University Press: Oxford University Press London, first edition) · Zbl 0020.29201
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