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

and

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.^{ [1] }

The first known proof is attributed to AxelThue ( 1902 ),^{ [2] } who used a pigeonhole argument.^{ [3] }^{ [4] } 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 *a*^{2} + 1 = 0 mod *p*. (Such an "a" is guaranteed for "p" by Wilson's Theorem.^{ [5] })

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*/*y*, 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

and

- ,

with

and

then

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.^{ [6] }

The proof is rather easy: by multiplying each congruence by the other *y _{i}* and subtracting, one gets

The hypotheses imply that each term has an absolute value lower than *XY* < *m*/2, 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.^{ [7] }

More precisely, given the two integers m and a appearing in Thue's lemma, the extended Euclidean algorithm computes three sequences of integers (*t*_{i}), (*x*_{i}) and (*y*_{i}) such that

where the *x*_{i} are non-negative and strictly decreasing. The desired solution is, up to the sign, the first pair (*x*_{i}, *y*_{i}) such that *x*_{i} < *X*.

- Padé approximant, a similar theory, for approximating Taylor series by rational functions

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**AN codes** are error-correcting code that are used in arithmetic applications. Arithmetic codes were commonly used in computer processors to ensure the accuracy of its arithmetic operations when electronics were more unreliable. Arithmetic codes help the processor to detect when an error is made and correct it. Without these codes, processors would be unreliable since any errors would go undetected. AN codes are arithmetic codes that are named for the integers and that are used to encode and decode the codewords.

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.

- Shoup, Victor (2005).
*A Computational Introduction to Number Theory and Algebra*(PDF). Cambridge University Press. Retrieved 26 February 2016.

- ↑ Shoup, theorem 2.33
- ↑ Thue, A. (1902), "Et par antydninger til en taltheoretisk metode",
*Kra. Vidensk. Selsk. Forh.*,**7**: 57–75 - ↑ Clark, Pete L. "Thue's Lemma and Binary Forms". CiteSeerX 10.1.1.151.636 .Cite journal requires
`|journal=`

(help) - ↑ Löndahl, Carl (2011-03-22). "Lecture on sums of squares" (PDF). Retrieved 26 February 2016.Cite journal requires
`|journal=`

(help) - ↑ Ore, Oystein (1948),
*Number Theory and its History*, pp. 262–263 - ↑ Shoup, section 4.6
- ↑ Shoup, section 4.5

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