Eisenstein integers

The Eisenstein integers, sometimes also called the Eisenstein-Jacobi integers, are numbers of the form a+bomega, where a and b are normal integers,
omegacongruent1/2(-1+isqrt(3))
is one of the roots of z^3 = 1, the others being 1 and
omega^2 = 1/2(-1-isqrt(3)).
The sums, differences, and products of Eisenstein integers is another Eisenstein integer.
Eisenstein integers are complex numbers that are members of the imaginary quadratic field Q(sqrt(-3)), which is precisely the ring Z[omega]. 

The field of Eisenstein integers has the six units (or roots of unity), namely ±1, ±omega, and ±omega^2.

Gaussian integer

A Gaussian integer is a complex number a+bi where a and b are integers. 

The Gaussian integers are members of the imaginary quadratic field Q(sqrt(-1)) and form a ring often denoted Z[i], or sometimes k(i). The sum, difference, and product of two Gaussian integers are Gaussian integers, but (a+bi)|(c+di) only if there is an e+fi such that
(a+bi)(e+fi) = (ae-bf)+(af+be) i = c+di
(Shanks 1993).
Gaussian integers can be uniquely factored in terms of other Gaussian integers (known as Gaussian primes) up to powers of i and rearrangements.
The units of Z[i] are ±1 and ±i.
One definition of the norm of a Gaussian integer is its complex modulus
|a+ib| = sqrt(a^2+b^2).

Gaussian primes

Gaussian primes are Gaussian integers z = a+bi satisfying one of the following properties. 
1. If both a and b are nonzero then, a+bi is a Gaussian prime iff a^2+b^2 is an ordinary prime. 
2. If a = 0, then bi is a Gaussian prime iff |b| is an ordinary prime and |b|congruent3 (mod 4). 
3. If b = 0, then a is a Gaussian prime iff |a| is an ordinary prime and |a|congruent3 (mod 4). 
The above plot of the complex plane shows the Gaussian primes as filled squares.
The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ....

 The Gaussian primes with |a|, |b|<=5 are given by -5-4i, -5-2i, -5+2i, -5+4i, -4-5i, -4-i, -4+i, -4+5i, -3-2i, -3, -3+2i, -2-5i, -2-3i, -2-i, -2+i, -2+3i, -2+5i, -1-4i, -1-2i, -1-i, -1+i, -1+2i, -1+4i, -3 i, 3i, 1-4i, 1-2i, 1-i, 1+i, 1+2i, 1+4i, 2-5i, 2-3i, 2-i, 2+i, 2+3i, 2+5i, 3-2i, 3, 3+2i, 4-5i, 4-i, 4+i, 4+5i, 5-4i, 5-2i, 5+2i, 5+4i.

Nombres premiers

Un nombre premier est un nombre possédant exactement deux diviseurs : 1 et lui-même.

A positive integer that has exactly one positive integer divisor other than 1 (i.e., no factors other than 1 and itself).  Prime numbers are often simply called primes.

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer pɭ that has no positive integer divisors other than 1 and p itself. 
(More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1.)
 For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24 = 2^3·3), making 24 not a prime number. 
Positive integers other than 1 which are not prime are called composite numbers.
Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers n whose divisors are trivial and given by exactly 1 and n.
While the term "prime number" commonly refers to prime positive integers,
 other types of primes are also defined, such as the Gaussian primes.


source WOLFRAM

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