Primes in residual classes

Primes of form a*n+d for fixed a and d. Also known as primes congruent to d modulo a.

Special cases:

2n+1 odd primes
4n+1 Pythagorean primes
4n+3 interger Gaussian primes

Any other special cases of this type that have been named?

[Batalov]

See http://en.wikipedia.org/wiki/Categor..._prime_numbers
and then google some more.

I resisted temptation to hyperlink the word google and/or add the Bart Simpson picture.

I have found several sites with classes of prime numbers. However I have not found any additional classes for the function a*n+d.

I found several sites listing classes of primes.

None listed additional classes using the function a*n +d.

[R.D. Silverman]

Originally Posted by Unregistered

Primes of form a*n+d for fixed a and d. Also known as primes congruent to d modulo a.

Special cases:

2n+1 odd primes
4n+1 Pythagorean primes
4n+3 interger Gaussian primes

Any other special cases of this type that have been named?

Primes of the form 4n+3 are not the Gaussian primes.

[cheesehead]

Originally Posted by Unregistered

4n+3 interger Gaussian primes

Expanding on Dr. Silverman's answer:

Gaussian primes are among the Gaussian integers. (http://en.wikipedia.org/wiki/Gaussian_prime) Gaussian integers are complex numbers a+bi. Gaussian primes have either:

A) a and b nonzero, and a² + b² is prime,

or

B) a is a prime of the form 4n+3 and b = 0,

or

C) a = 0 and b is a prime of the form 4n+3.

So, case B) Gaussian primes have values equal to real (i.e., imaginary part = 0) integer primes, and some folks may (sloppily) write as though those were the only Gaussian primes. However, use of the adjective Gaussian really should imply knowledge of their complex nature and that not all Gaussian primes are real integer primes.

Perhaps I misunderstand the term interger. I thought that that indicating these were integers implied that the imaginary part must be zero.