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A224807 (-1)^((p-1)/2)*Binomial(p-1,(p-1)/2) mod p^3 where p is the n-th prime. 0


%S 25,6,323,1079,924,3044,6252,254,21084,4217,42514,48955,63168,101333,

%T 90896,87970,164396,100099,85982,221337,464837,90637,214936,735552,

%U 171600,330425,437835,311632,363522

%N (-1)^((p-1)/2)*Binomial(p-1,(p-1)/2) mod p^3 where p is the n-th prime.

%C This sequence is related to Morley's Congruence which states that, for prime p>2, (-1)^((p-1)/2)*binomial(p-1,(p-1)/2) == 4^(p-1) (mod p^3).

%C It of interest to note that this congruence can only be illustrated in Maple by using the right hand side of the identity a== b (mod m) iff m|(a-b). Checking for values of n^3 that divide ((-1)^((n-1)/2)*binomial(n-1,(n-1)/2) - 4^(n-1)) produces the sequence of primes. Encoding the left hand side produces this sequence.

%C a(n) == 1 (mod p)

%p p:= n-> ithprime(n): seq((-1)^((p(n)-1)/2)*binomial(p(n)-1,(p(n)-1)/2) mod p(n)^3, n=2..30)

%K nonn

%O 2,1

%A _Gary Detlefs_, Apr 18 2013

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Last modified January 26 23:54 EST 2022. Contains 350601 sequences. (Running on oeis4.)