We use results from the theory of algebras with polynomial identities (PI algebras) to study the witness complexity of matrix identities. A matrix identity of d× d matrices over a field F is a non-commutative polynomial f(x1, . . . , xn) over F, such that f vanishes on every d×d matrix assignment to its variables. For any field F of characteristic 0, any d > 2 and any finite basis of d× d matri...

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities a(bc) = a(cb), (ba)a = (ba)a, (b, a, c) = 2(b, a, c)a. These identities are satisfied by the product ab = a a b + b ` a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤ 7 is a consequence of the three identities in degree ≤ 4, bu...

The Cunningham project seeks to factor numbers of the form bn±1 with b = 2, 3, . . . small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing bn by a polynomial in such a way that polynomial factorization is possible. For example, by substituting y = 2k into the polynomial factorization (2y2)2+1 = (2y2−2y+1)(2y2+2y+1) we ca...