1,2

Denote T(p) = binomial(4p-1, 2p-1) mod p^5, where p is the n-th prime. Theorem 30 in the link below states that T(p) = binomial(4p, p) - 1 for p > 5. This is difficult to empirically demonstrate as T(n) = 3, 219, 1753, 7549... <> binomial(4p, p) - 1 (binomial(4p-1, 2p-1) - binomial(4p, p)+1)/p^5 = 27/32, 44/27, 87533/3125, 19681560/16807...not integer.

Thus the identity seems to violate both the left and right hand sides of the identity a == b (mod m) if and only if m|(a-b).

It is of interest to note however that T(p) mod p = 3 for p > 3 and that T(p) - 3 is divisible by p^3 (this sequence).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

p:= ithprime: T:= n-> binomial(4*p(n)-1, 2*p(n)-1) mod p(n)^5: seq((T(n)-3)/p(n)^3, n=1..40);

a[n_] := Module[{p = Prime[n]}, (Mod[Binomial[4p-1, 2p-1], p^5]-3)/p^3]; Array[a, 40] (* Jean-François Alcover, Feb 16 2019 *)

Sequence in context: A063216 A238290 A100315 * A248700 A001049 A134445

Adjacent sequences: A224949 A224950 A224951 * A224953 A224954 A224955

Gary Detlefs, Apr 20 2013

approved