mersenneforum.org Primes in residual classes
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 2008-09-05, 19:59 #1 Unregistered   130010 Posts 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?
 2008-09-06, 03:43 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·3,229 Posts 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.
 2008-09-07, 15:45 #3 Unregistered   8,779 Posts Thank you Batalov I have found several sites with classes of prime numbers. However I have not found any additional classes for the function a*n+d.
 2008-09-07, 16:43 #4 Unregistered   22·5·11·31 Posts Thank you Batalov I found several sites listing classes of primes. None listed additional classes using the function a*n +d.
2008-09-08, 00:32   #5
R.D. Silverman

Nov 2003

164448 Posts

Quote:
 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.

2008-09-09, 06:26   #6

"Richard B. Woods"
Aug 2002
Wisconsin USA

1E0C16 Posts

Quote:
 Originally Posted by Unregistered 4n+3 interger Gaussian primes

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.

Last fiddled with by cheesehead on 2008-09-09 at 06:28

 2008-09-11, 12:57 #7 Unregistered   24×277 Posts Gaussian interger primes Perhaps I misunderstand the term interger. I thought that that indicating these were integers implied that the imaginary part must be zero.

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