Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Georg Cantor's first set theory article wikipedia , lookup
List of first-order theories wikipedia , lookup
Mathematical proof wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Elementary mathematics wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Collatz conjecture wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Congruences One of the important notational devices used by Gauss in his Disquisitiones Arithmeticae (1801) was the congruence: where a, b, m are integers and m is nonzero, he writes a ≡ b (modm), read as “a is congruent to b modulo m”, to mean that a ≡ b (modm) ⇔ m|(a – b) ⇔ € same remainder when divided by m a,b have the Here, m is called the modulus. Congruences are € € prototypical examples of equivalence relations: € Proposition Congruence mod m is an equivalence relation (it is reflexive, symmetric, transitive). // At least as important is the fact that congruence mod m is compatible with arithmetic. Proposition If a ≡ b (modm) and c ≡ d (mod m), then (1) a + c ≡ b + d (mod m); (2) ac ≡ bd (mod m); € € (3) a k ≡ b k (mod m) for any positive integer k. // € € € Proposition (1) Reduction: If a ≡ b (modm) and n|m, then a ≡ b (modn). m ). (2) Cancellation: ac ≡ bc (mod m) ⇒ a ≡ b (mod (c,m ) € € € € Because congruence mod m is an equivalence relation, Z is partitioned into equivalence classes under this relation, called more appropriately congruence classes mod m. (Thus, every integer belongs to exactly one congruence class mod m and no two congruence classes have any numbers in common.) There are exactly m congruence classes mod m and they are determined by the m possible remainders (or in Gauss’ terminology, residues) r = 0, 1, … , m – 1 on division by m. These m numbers constitute the standard residue system (SRS) mod m, e.g. {0, 1, 2, 3, 4, 5, 6} is a SRS mod 7. Replacing anyone of the residues by any number to which it is congruent yields another complete residue system (CRS), e.g., {7, 50, 30, 3, –3, 5, –1} is a CRS mod 7. A least absolute residue system mod m is a CRS whose members have the smallest absolute values possible, e.g., {–3, –2, –1, 0, 1, 2, 3} is a least absolute residue system mod 7. Here is an interesting application of congruence arithmetic: Theorem There are infinitely many primes of the form p ≡ −1(mod 4). Proof We argue in a manner similar to Euclid’s proof that there are infinitely many primes. € Suppose there are only finitely many primes of the desired form. List them: q1 = 3,q 2 = 7,… ,qn . Now consider the number N = 4 q2 qn + 3, which is certainly larger than all of the q’s and divisible by none of them. If the prime factorization of N is € € m N = ∏ p kek , k=1 then since N must be odd, all the p’s are odd. So it must be that each pk ≡ 1(mod 4). But then on the € one hand, m €N = ∏ k=1 pkek m ≡ ∏ 1e k ≡ 1(mod 4), k=1 while on the other, N = 4 q2 qn + 3 ≡ −1(mod 4). Thus, 1 ≡ −1 (mod 4). This contradiction completes € the proof. // € €