Fermat's Little Theorem

If  is a prime number and  is a natural number, then

(1)

Furthermore, if  ( does not divide ), then there exists some smallest exponent  such that

(2)

and  divides . Hence,

(3)

The theorem is sometimes also simply known as "Fermat's theorem" (Hardy and Wright 1979, p. 63).

This is a generalization of the Chinese hypothesis and a special case of Euler's totient theorem. It is sometimes called Fermat's primality test and is a necessary but not sufficient test for primality. Although it was presumably proved (but suppressed) by Fermat, the first proof was published by Euler in 1749. It is unclear when the term "Fermat's little theorem" was first used to describe the theorem, but it was used in a German textbook by Hensel (1913) and appears in Mac Lane (1940) and Kaplansky (1945).

The theorem is easily proved using mathematical induction on . Suppose  (i.e.,  divides ). Then examine

(4)

From the binomial theorem,

(5)

Rewriting,

(6)

But  divides the right side, so it also divides the left side. Combining with the induction hypothesis gives that  divides the sum

(7)

as assumed, so the hypothesis is true for any . The theorem is sometimes called Fermat's simple theorem. Wilson's theorem follows as a corollary of Fermat's little theorem.

Fermat's little theorem shows that, if  is prime, there does not exist a base  with  such that  possesses a nonzero residue modulo . If such base  exists,  is therefore guaranteed to be composite. However, the lack of a nonzero residue in Fermat's little theorem does not guarantee that  is prime. The property of unambiguously certifying composite numbers while passing some primes make Fermat's little theorem a compositeness test which is sometimes called the Fermat compositeness test. A number satisfying Fermat's little theorem for some nontrivial base and which is not known to be composite is called a probable prime.

Composite numbers known as Fermat pseudoprimes (or sometimes simply "pseudoprimes") have zero residue for some s and so are not identified as composite. Worse still, there exist numbers known as Carmichael numbers (the smallest of which is 561) which give zero residue for any choice of the base  relatively prime to . However, Fermat's little theorem converse provides a criterion for certifying the primality of a number. A table of the smallest pseudoprimes  for the first 100 bases  follows (OEIS A007535; Beiler 1966, p. 42 with typos corrected).

2	341	22	69	42	205	62	63	82	91
3	91	23	33	43	77	63	341	83	105
4	15	24	25	44	45	64	65	84	85
5	124	25	28	45	76	65	112	85	129
6	35	26	27	46	133	66	91	86	87
7	25	27	65	47	65	67	85	87	91
8	9	28	45	48	49	68	69	88	91
9	28	29	35	49	66	69	85	89	99
10	33	30	49	50	51	70	169	90	91
11	15	31	49	51	65	71	105	91	115
12	65	32	33	52	85	72	85	92	93
13	21	33	85	53	65	73	111	93	301
14	15	34	35	54	55	74	75	94	95
15	341	35	51	55	63	75	91	95	141
16	51	36	91	56	57	76	77	96	133
17	45	37	45	57	65	77	247	97	105
18	25	38	39	58	133	78	341	98	99
19	45	39	95	59	87	79	91	99	145
20	21	40	91	60	341	80	81	100	153
21	55	41	105	61	91	81	85

REFERENCES:

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 61, 1987.

Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 141-142, 1996.

Courant, R. and Robbins, H. "Fermat's Theorem." §2.2 in Supplement to Ch. 1 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 37-38, 1996.

Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, pp. 118-125, 2000.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Hensel, K. Zahlentheorie. Berlin: G. J. Göschen, 1913.

Kaplansky, I. "Lucas's Tests for Mersenne Numbers." Amer. Math. Monthly 52, 188-190, 1945.

Mac Lane, S. "Modular Fields." Amer. Math. Monthly 47, 259-274, 1940.

Nagell, T. "Fermat's Theorem and Its Generalization by Euler." §21 in Introduction to Number Theory. New York: Wiley, pp. 71-73, 1951.

Séroul, R. "The Theorems of Fermat and Euler." §2.8 in Programming for Mathematicians. Berlin: Springer-Verlag, p. 15, 2000.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 20, 1993.

Sloane, N. J. A. Sequence A007535/M5440 in "The On-Line Encyclopedia of Integer Sequences."