Primality proof for n = 76884956397045344220809746629001649093037950200943055203735601445031516197751:

Take b = 2.

b^(n-1) mod n = 1.

3059213862715144055733503214373292934438943635608167530247 is prime.
b^((n-1)/3059213862715144055733503214373292934438943635608167530247)-1 mod n = 35407069651394190740765184689298898599787371795830213440776353901932813443171, which is a unit, inverse 32451835672024130223757551332131052571135022961631005019534398669207120274534.

(3059213862715144055733503214373292934438943635608167530247) divides n-1.

(3059213862715144055733503214373292934438943635608167530247)^2 > n.

n is prime by Pocklington's theorem.