Industrial-grade primes (the term is apparently due to ) are for which has not been certified (i.e. rigorously proven), but they have undergone tests such as the , which has a positive, but negligible, failure rate, or the , which no composites are known to pass.
Industrial-grade primes are sometimes used instead of certified primes in such as , which require the user to generate large . of large numbers (over 100 digits for instance) is significantly harder than showing they are industrial-grade primes. The latter can be done almost instantly with a so low that it is highly unlikely to ever fail in practice. In other words, the number is believed to be prime with very high, but not absolute, confidence.