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Old 2008-09-05, 19:59   #1
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Default 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?
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Old 2008-09-06, 03:43   #2
Batalov
 
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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.
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Old 2008-09-07, 15:45   #3
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Smile 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.
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Old 2008-09-07, 16:43   #4
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Smile Thank you Batalov

I found several sites listing classes of primes.

None listed additional classes using the function a*n +d.
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Old 2008-09-08, 00:32   #5
R.D. Silverman
 
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Quote:
Originally Posted by Unregistered View Post
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.
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Old 2008-09-09, 06:26   #6
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Quote:
Originally Posted by Unregistered View Post
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.

Last fiddled with by cheesehead on 2008-09-09 at 06:28
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Old 2008-09-11, 12:57   #7
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Question 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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