In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus.
More precisely for every pair of integers (a, m) with m > 1, given two positive integers X and Y such that X ≤ m < XY, there are two integers x and y such that
Usually, one takes X and Y equal to the smallest integer greater than the square root of m, but the general form is sometimes useful, and make the unicity theorem (below) easier to state.
The first known proof is attributed to AxelThue ( 1902 ), who used a pigeonhole argument. It can be used to prove Fermat's theorem on sums of two squares by taking m to be a prime p that is 1 mod 4 and taking a to satisfy a2 + 1 = 0 mod p. (Such an "a" is guaranteed for "p" by Wilson's Theorem. )
In general, the solution whose existence is asserted by Thue's lemma is not unique. For example, when a = 1 there are usually several solutions (x,y) = (1,1), (2,2), (3,3), ..., providing that X and Y are not too small. Therefore, one may only hope unicity for the rational number x/, to which a is congruent modulo m if y and m are coprime. Nevertheless, this rational number needs not to be unique; for example, if m = 5, a = 2 and X = Y = 3, one has the two solutions
However, for X and Y small enough, if a solution exists, it is unique. More precisely, with above notation, if
This result is the basis for rational reconstruction, which allows using modular arithmetic for computing rational numbers for which one knows bounds for numerators and denominators.
The proof is rather easy: by multiplying each congruence by the other yi and subtracting, one gets
The hypotheses imply that each term has an absolute value lower than XY < m/, and thus that the absolute value of their difference is lower than m. This implies that , and thus the result.
The original proof of Thue's lemma is not efficient, in the sense that it does not provide any fast method for computing the solution. The extended Euclidean algorithm, allows us to provide a proof that leads to an efficient algorithm that has the same computational complexity of the Euclidean algorithm.
More precisely, given the two integers m and a appearing in Thue's lemma, the extended Euclidean algorithm computes three sequences of integers (ti), (xi) and (yi) such that
where the xi are non-negative and strictly decreasing. The desired solution is, up to the sign, the first pair (xi, yi) such that xi < X.
In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:
Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
In number theory, Euler's theorem states that if n and a are coprime positive integers, then
In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x2 – 4.
This article collects together a variety of proofs of Fermat's little theorem, which states that
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.
In arithmetic and algebra, the cube of a number n is its third power: the result of the number multiplied by itself twice:
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4 ,1/5, etc.
Modular exponentiation is a type of exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography.
In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication. It was introduced in 1985 by the American mathematician Peter L. Montgomery.
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:
In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.
In mathematics, in particular the area of number theory, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as
Pocklington's algorithm is a technique for solving a congruence of the form
In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation , where and d and m are coprime. The algorithm was described in 1908 by Giuseppe Cornacchia.
Coppersmith's attack describes a class of cryptographic attacks on the public-key cryptosystem RSA based on the Coppersmith method. Particular applications of the Coppersmith method for attacking RSA include cases when the public exponent e is small or when partial knowledge of the secret key is available.
In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by Eisenstein (1850), though Jacobi had previously announced a similar result for the special cases of 5th, 8th and 12th powers in 1839.