Timeline for Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$

12 events

when toggle format	what		by	license	comment
Dec 24, 2015 at 2:08	history	edited	Chitsai Liu	CC BY-SA 3.0	deleted 77 characters in body
Dec 23, 2015 at 11:32	vote	accept	Chitsai Liu
Dec 23, 2015 at 10:49	comment	added	Ilya Bogdanov		Isn't it true that simply $S((p-1)/2)$ is divisible by $p^2$ for $p\geq 7$? It seems quite plausible, and this would reveal the magic about the coefficients 2 and 7 which are needed in this case only for $p=5$...
Dec 23, 2015 at 7:14	answer	added	Ofir Gorodetsky		timeline score: 20
Dec 23, 2015 at 6:03	history	edited	Chitsai Liu	CC BY-SA 3.0	added 4 characters in body
Dec 23, 2015 at 5:56	history	edited	Chitsai Liu	CC BY-SA 3.0	added 257 characters in body
Dec 23, 2015 at 5:48	comment	added	Chitsai Liu		@ tkr, I have checked your congruence mod $p^3$ for $p\ge 7$ and find it is true, which I didn't notice. Thanks!
Dec 23, 2015 at 4:57	comment	added	tkr		I don't know if it is helpful, and maybe you noticed, but for primes $p > 5$ it seems (unless my code is wrong) that that your second congruence in the "comment" is true modulo $p^3$ (I checked all $p < 10000$, again unless my code wrong).
Dec 23, 2015 at 4:44	history	edited	Chitsai Liu	CC BY-SA 3.0	added 362 characters in body
Dec 23, 2015 at 1:04	history	edited	Chitsai Liu	CC BY-SA 3.0	edited title
Dec 23, 2015 at 0:49	history	edited	Chitsai Liu	CC BY-SA 3.0	edited title
Dec 23, 2015 at 0:41	history	asked	Chitsai Liu	CC BY-SA 3.0