Fermat Pseudoprime

A Fermat pseudoprime to a base , written psp(), is a composite number  such that , i.e., it satisfies Fermat's little theorem. Sometimes the requirement that  must be odd is added (Pomerance et al. 1980) which, for example would exclude 4 from being considered a psp(5).

psp(2)s are called Poulet numbers or, less commonly, Sarrus numbers or Fermatians (Shanks 1993). The following table gives the first few Fermat pseudoprimes to some small bases .

	OEIS	-Fermat pseudoprimes
2	A001567	341, 561, 645, 1105, 1387, 1729, 1905, ...
3	A005935	91, 121, 286, 671, 703, 949, 1105, 1541, 1729, ...
4	A020136	15, 85, 91, 341, 435, 451, 561, 645, 703, ...
5	A005936	4, 124, 217, 561, 781, 1541, 1729, 1891, ...

If base 3 is used in addition to base 2 to weed out potential composite numbers, only 4709 composite numbers remain . Adding base 5 leaves 2552, and base 7 leaves only 1770 composite numbers.

The following table gives the number of Fermat pseudoprimes to various small bases less than 10, , , ....

base(s)	OEIS	Fermat pseudoprimes less than 10, , ...
2	A055550	0, 0, 3, 22, 78, 245, 750, 2057, ...
2, 3	A114246	0, 0, 0, 7, 23, 66, 187, 485, ...
2, 3, 5	A114248	0, 0, 0, 4, 11, 36, 95, 257, ...
2, 3, 5, 7	A114250	0, 0, 0, 0, 3, 19, 63, 175, ...
3	A114245	0, 1, 6, 23, 78, 246, 760, 2155, ...
5	A114247	1, 1, 5, 20, 73, 248, 745, 1954, ...
7	A114249	1, 2, 6, 16, 73, 234, 659, 1797, ...

REFERENCES:

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 182, 1998.

Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimes to ." Math. Comput. 35, 1003-1026, 1980. https://mpqs.free.fr/ThePseudoprimesTo25e9.pdf.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 115, 1993.

Sloane, N. J. A. Sequences A001567/M5441, A005935/M5362, A005936/M3712, A020136, A055550, A114245, A114246, A114247, A114248, A114249, and A114250 in "The On-Line Encyclopedia of Integer Sequences."