For this post I will only count the operations PLUS, MINUS, MULT. They may be done on rather large numbers.
Recall that from the work coming out of Hilberts 10th problem we know the following: For every c.e. set (used to be called r.e., some people still do) there is a polynomial f in 13 or less variables (we'll assume 13) with coefficients in the integers such that
x in A iff (∃ a1,...,a13))[f(x,a1,...,a13)=0]
In an article about Hilbert's 10th problem written in 1974 by Davis-Matiyasevich-Robinson they note that by this result there is a FINITE number M such that, for ALL primes p, there is a certification that p is prime that uses at most M operations: given p a prime let a1,...,a13 be such that f(p,a1,...,a13)=0. The certification that p is prime is just the evaluation of that polynomial and seeing that its 0.
Is this still the only proof that one can certify primality in a CONSTANT number of operations?
Primes is irrelevant to all of this--- any c.e. set would work. (The result for c.e. sets may qualify as a theorem that is less interesting because its more interesting.) But for primes I am wondering if there is another way to do this- perhaps using number theory, perhaps with a smaller value of M. For the explicit poly for primes, due to Jones, see here.