A recursive primality certificate for a prime . The certificate consists of a list of
1. A point on an elliptic curve
for some numbers and .
2. A prime with , such that for some other number and with , is the identity on the curve, but is not the identity. This guarantees primality of by a theorem of Goldwasser and Kilian (1986).
3. Each has its recursive certificate following it. So if the smallest is known to be prime, all the numbers are certified prime up the chain.
A Pratt certificate is quicker to generate for small numbers. The Wolfram Language task ProvablePrimeQ[n] in the Wolfram Language package PrimalityProving` therefore generates an Atkin-Goldwasser-Kilian-Morain certificate only for numbers above a certain limit ( by default), and a Pratt certificate for smaller numbers.