Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as $Z [i]$ .^[1] This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.

Gaussian integers as lattice points in the complex plane

Formal definition

Formally, the Gaussian integers are the set

\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.

^[1]

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the $2$ -dimensional integer lattice.

Norm of a Gaussian integer

The (arithmetic or field) norm of a Gaussian integer is the square of its absolute value (Euclidean norm) as a complex number. It is the natural number defined as

N(a+bi) = a^2+b^2 = (a+bi)\overline{(a+bi)} = (a+bi)(a-bi),

where $\cdot$ (an overline) is complex conjugation.

The norm is multiplicative, since the absolute value of complex numbers is multiplicative, i.e., one has

N(zw) = N(z)N(w).

^[2]

The latter can also be verified by a straightforward check. The units of $Z [i]$ are precisely those elements with norm $1$ , i.e. the set ${\pm1, \pm i}.$ ^[3]

As a principal ideal domain

The Gaussian integers form a principal ideal domain with units ${\pm1, \pm i}.$ For $x \in Z [i]$ , the four numbers $\pm x, \pm ix$ are called the associates of $x$ . As for every principal ideal domain, $Z [i]$ is also a unique factorization domain.

The prime elements of $Z [i]$ are also known as Gaussian primes. An associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes. The positive integer Gaussian primes are the prime numbers that are congruent to $3$ modulo $4$ , (sequence A002145 in OEIS). One should not refer to only these numbers as "the Gaussian primes", which refers to all the Gaussian primes, many of which do not lie in $Z$ .^[4]

A Gaussian integer $a + bi$ is a Gaussian prime if and only if either:

one of $a, b$ is zero and the other is a prime number of the form $4 n + 3$ (with $n$ a nonnegative integer) or its negative $-(4 n + 3)$ , or
both are nonzero and $a 2 + b 2$ is a prime number (which will not be of the form $4 n + 3$ ).

In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, or it is the product by a unit ( $\pm1, \pm i$ ) of a prime number of the form $4 n + 3$ .

It follows that there are three cases for the factorization of a prime number $p$ in the Gaussian integers:

If $p$ is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of algebraic number theory, $p$ is said to be inert in the Gaussian integers.
If $p$ is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); $p$ is said to be a decomposed prime in the Gaussian integers. For example, $5 = (2+ i)\cdot(2- i)$ and $13 = (3+2 i)\cdot(3-2 i)$ .
If $p = 2$ , we have $2 = (1 + i)(1 - i) = i (1 - i) 2$ ; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique ramified prime in the Gaussian integers.

As an integral closure

The ring of Gaussian integers is the integral closure of $Z$ in the field of Gaussian rationals $Q (i)$ consisting of the complex numbers whose real and imaginary part are both rational.

As a Euclidean domain

It is easy to see graphically that every complex number is within $\tfrac{\sqrt 2}{2}$ units of a Gaussian integer.

Put another way, every complex number (and hence every Gaussian integer) has a maximal distance of

\frac{\sqrt 2}{2}\sqrt{N(z)}

units to some multiple of $z$ , where $z$ is any Gaussian integer; this turns $Z [i]$ into a Euclidean domain, where

v(z) = N(z).

^[5]

Historical background

The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832) (see ). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence $x 2 \equiv q (mod p)$ to that of $x 2 \equiv p (mod q)$ . Similarly, cubic reciprocity relates the solvability of $x 3 \equiv q (mod p)$ to that of $x 3 \equiv p (mod q)$ , and biquadratic (or quartic) reciprocity is a relation between $x 4 \equiv q (mod p)$ and $x 4 \equiv p (mod q)$ . Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).

In a footnote he notes that the Eisenstein integers are the natural domain for stating and proving results on cubic reciprocity and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws.

This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.

Unsolved problems

Repartition in the plane of the small Gaussian primes

Most of the unsolved problems are related to the repartition in the plane of the Gaussian primes.

Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.

There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:

The real and imaginary axes have the infinite set of Gaussian primes 3, 7, 11, 19, ... and their associates. Are there any other lines that have infinitely many Gaussian primes on them? In particular, are there infinitely many Gaussian primes of the form 1+ki?^[6]
Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of bounded length? This is known as the Gaussian moat problem; it was posed in 1962 by Basil Gordon and remains unsolved.^[7]^[8]

Notes

1 2 Fraleigh (1976, p. 286)
↑ Fraleigh (1976, p. 289)
↑ Fraleigh (1976, p. 288)
↑ , OEIS sequence A002145 "COMMENT" section
↑ Fraleigh (1976, p. 287)
↑ Ribenboim, Ch.III.4.D Ch. 6.II, Ch. 6.IV (Hardy & Littlewood's conjecture E and F)
↑ Gethner, Ellen; Wagon, Stan; Wick, Brian (1998). "A stroll through the Gaussian primes". The American Mathematical Monthly 105 (4): 327–337. doi:10.2307/2589708. MR 1614871. Zbl 0946.11002.
↑ Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 55–57. ISBN 978-0-387-20860-2. Zbl 1058.11001.

References

C. F. Gauss, Theoria residuorum biquadraticorum. Commentatio secunda., Comm. Soc. Reg. Sci. Göttingen 7 (1832) 1-34; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93–148.
Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
Kleiner, Israel (1998). "From Numbers to Rings: The Early History of Ring Theory". Elem. Math. 53 (1): 18–35. doi:10.1007/s000170050029. Zbl 0908.16001.
Ribenboim, Paulo (1996). The New Book of Prime Number Records (3rd ed.). New York: Springer. ISBN 0-387-94457-5. Zbl 0856.11001.

External links

Henry G. Baker (1993) Complex Gaussian Integers for 'Gaussian Graphics', ACM SIGPLAN Notices, Vol. 28, Issue 11. DOI 10.1145/165564.165571 (html)
IMO Compendium text on quadratic extensions and Gaussian Integers in problem solving
Weisstein, Eric W., "Landau's Problems", MathWorld.

Algebraic numbers

Algebraic integer Chebyshev nodes Constructible number Conway's constant ³√2 Eisenstein integer Gaussian integer $φ$ Perron number Pisot–Vijayaraghavan number Quadratic irrational ℚ Root of unity Salem number $δ$ _$S$ √2 √3 √5 ¹²√2		ℚ

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 100 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541

List of prime numbers

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