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Fermat’s little theorem has big consequences in number theory and its applications, such as data encryption.[1] It is called “little” only in deference to Fermat’s last theorem, and some authors in fact call it just Fermat’s theorem.[2]
Theorem. (Pierre de Fermat)
Given a prime number and an integer coprime to , .
Proof. Consider the numbers | b, 2 b, 3 b, ..., ( p − 1) b |
. Those are all different and nonzero , since is prime and is coprime to . Indeed, if , then , since we can “cancel the ” ( being coprime to ). So we have numbers, all different, and none of them is 0 . So they must be congruent to in some order. Multiplying them together, we get . So their product is congruent to .
We now have two expressions for this product, so we can equate them: | ( p − 1)! b p − 1 ≡ ( p − 1)! (mod p). |
Now is coprime to , so we can again cancel, to give . □
For example, 10 12 is 1 more than 999999999999, which is divisible by 13 (as it is 76923076923 × 13).
Fermat’s little theorem is a special case of
Euler’s theorem that
if
,
[3] since
and
for any
not a multiple of
.
The theorem is best demonstrated with small odd primes, such as
3 and
5. In the case of
, the theorem tells us that no square is one less than a multiple of
3. If we take integers that are not multiples of
3, i.e
{1, 2, 4, 5, 7, 8, 10, 11, ...}, square them
{1, 4, 16, 25, 49, 64, 100, 121, ...}, and subtract
1 from each, we get multiples of
3 {0, 3, 15, 24, 48, 63, 99, 120, ...} But it also holds for
, though giving as it does the uninteresting result that an odd number is not even number.
Fermat pseudoprimes
However, the theorem can’t be used as a
proof of primality for
, at least not if we only
test for one base
coprime to
. There are composite numbers
for which the
congruence holds for some bases
coprime to
.
For example, for the composite number 341 = 11 × 31, we have
- 2 340 = 2239744742177804210557442280568444278121645497234649534899989100963791871180160945380877493271607115776,
which leaves a remainder of 1 when divided by 341. However,
- 3 340 = 1664280806589814803858571371708626691451909331385734291010900950997276297957762658553727546535190828834204613885667545045874010453464713005017905547836267732294801,
and that leaves a remainder of
56 when divided by
341. These numbers are called
Fermat pseudoprimes to a given base
(the Fermat pseudoprimes to base
2 are also called
Poulet numbers). Though there are infinitely many such pseudoprimes, they “are sparsely distributed,” enabling the theorem to be used as a reasonable step in primality testing.
[4]
In the case of 341, we can see that among small bases, remainders other than 1 are given often enough, e.g.
A206786 Remainder of
divided by
341.
-
{1, 1, 56, 1, 67, 56, 56, 1, 67, 67, 253, 56, 67, 56, 1, 1, 56, 67, 56, 67, 67, 253, 1, 56, 56, 67, 1, 56, 1, 1, 155, 1, 187, 56, 1, 67, 56, 56, 1, 67, 67, 67, 56, 253, 56, 1, 1, 56, 67, 56, 67, 67, 67, 1, 242, ...}
Absolute Fermat pseudoprimes
So, what about testing for all bases
coprime to
? Unfortunately, some composite numbers are pseudoprimes to all bases
coprime to
, these are called
Carmichael numbers (or absolute Fermat pseudoprimes), the first such number being
561 = 3 × 11 × 17.
See also
Notes
- ↑ Schroeder (2009), p. 139.
- ↑ Such as Manfred R. Schroeder.
- ↑ Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007) p. 16, Theorems 9.6 and 9.7.
- ↑ R. Crandall and C. Pomerance. Prime Numbers: A Computational Perspective New York: Springer-Verlag (2001): p. 121.
References
- Manfred R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity, 5th Ed. Springer (2009).
External links
