Industrial-grade prime

Industrial-grade primes (the term is apparently due to Henri Cohen[1]) are integers for which primality has not been certified (i.e. rigorously proven), but they have undergone probable prime tests such as the Miller-Rabin primality test, which has a positive, but negligible, failure rate, or the Baillie-PSW primality test, which no composites are known to pass.

Industrial-grade primes are sometimes used instead of certified primes in algorithms such as RSA encryption, which require the user to generate large prime numbers. Certifying the primality 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 failure rate 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.
References

Chris Caldwell, The Prime Glossary: probable prime at The Prime Pages.

Prime number classes
By formula

Fermat (22n + 1) Mersenne (2p − 1) Double Mersenne (22p−1 − 1) Wagstaff (2p + 1)/3 Proth (k·2n + 1) Factorial (n! ± 1) Primorial (pn# ± 1) Euclid (pn# + 1) Pythagorean (4n + 1) Pierpont (2m·3n + 1) Quartan (x4 + y4) Solinas (2m ± 2n ± 1) Cullen (n·2n + 1) Woodall (n·2n − 1) Cuban (x3 − y3)/(x − y) Carol (2n − 1)2 − 2 Kynea (2n + 1)2 − 2 Leyland (xy + yx) Thabit (3·2n − 1) Williams ((b−1)·bn − 1) Mills (⌊A3n⌋)

By integer sequence

Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property

Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent

Happy Dihedral Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic

Patterns

Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain/Safe (p, 2p + 1) Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) Balanced (consecutive p − n, p, p + n)

By size

Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known

Complex numbers

Eisenstein prime Gaussian prime

Composite numbers

Pseudoprime
Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious