In mathematics, a strong pseudoprime is a certain kind of natural number. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

Formal definition

Formally, a natural number n := d · 2^s + 1 is called a strong pseudoprime to base a iff one of the following conditions hold: IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

It should be noted, however, that the definition is occasionally tightened so that only the first condition is sufficient for a number to be a strong pseudoprime.

Properties of strong pseudoprimes

A strong pseudoprime to base a is always an Euler pseudoprime to base a (Pomerance, Selfridge, Wagstaff 1980), but not all Euler pseudoprimes are strong pseudoprimes. Some Fermat pseudoprimes and Carmichael numbers are also strong pseudoprimes. An odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and (where mod refers to the modulo operation). ... An odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and (where mod refers to the modulo operation). ... A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime. ... In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn âˆ’ 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). ...

As Monier and Rabin showed in 1980, a composite number n is a strong pseudoprime to at most one quarter of all bases <n; thus, there are no "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases. 1980 (MCMLXXX) was a leap year starting on Tuesday. ...

It can, however, be proven (though not easily) that there are infinitely many strong pseudoprimes to any base.

Examples

The first few strong pseudoprimes to base 2 are 2047, 3277, 4033, 4681, 8321, 15841, 29341, ... (sequence A001262 in OEIS); the first few strong pseudoprimes to base 3 are 121, 703, 1891, 3281, 8401, 8911, 10585, ... (sequence A020229 in OEIS). The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

References

• C. Pomerance, J. L. Selfridge and Wagstaff, Jr., S. S., The pseudoprimes to 25 · 10⁹, Math. Comp., 35:151 (1980) 1003-1026