Primes Produced by Quadratic Forms

[Artur]

Because Sqrt[-12]=2Sqrt[-3] that case wasn't the best
A139490(2) is better: x^2+9y^2 and x^2+4xy+y^2
Discriminant x^2+9y^2, d= 0^2-4*9=-36  Sqrt[d]=6Sqrt[-1]
Discriminant x^2+4xy+y^2, d= 4^2-4*1=-12  Sqrt[d]=2Sqrt[-3]
Sets of primes both these are also that same
Artur


Artur pisze:
> Some sequences in ONEIS follow book of Cox (and other authors) 
> suggested and prooved that in theory of primes of binary forms 
> ax^2+bxy+cy^2 discriminant b^2-4ac is the most important.
> If we take first sample A139490 
> <http://www.research.att.com/%7Enjas/sequences/A139490>(1)=A007645 
> <http://www.research.att.com/%7Enjas/sequences/A007645>
> Discriminant of x^2 + xy + y^2 is 1^2-4*1*1= -3
> but discriminant of x^2 + 3*y^2=0^2-4*1*3= -12
> Sets of primes of both are that same
> Artur
>
>
> T. D. Noe pisze:
>>> Thus, every prime p such that (-30/p)=1 is produced by one of the
>>> quadratic forms x^2 + 30y^2, 2x^2 + 15y^2, 3x^2 + 10y^2 or 5x^2 + 
>>> 6y^2. It
>>> is easy to see (using quadratic reciprocity and its friends) that the
>>> primes p such that (-30/p)=1 are the primes where p == 1, 11, 13, 
>>> 17, 23,
>>> 29, 31, 37, 43, 47, 49, 59, 67, 79, 101, or 113 (mod 120).
>>>
>>> The first form can, looking modulo 120, only produce primes p == 1, 
>>> 31, 49
>>> or 79 (mod 120).
>>> The second form can, looking modulo 120, only produce primes p == 
>>> 17, 47,
>>> 113 or 23 (mod 120).
>>> The third form can, looking modulo 120, only produce primes p == 13, 
>>> 43,
>>> 37 or 67 (mod 120).
>>> The fourth form can, looking modulo 120, only produce primes p == 
>>> 11, 101,
>>> 59 or 29 (mod 120).
>>>
>>> Since each congruence class modulo 120 of primes p such that 
>>> (-30/p)=1 is
>>> represented only once above, it follows that those congruential 
>>> conditions
>>> are not only necessary but sufficient for expressibility in any of 
>>> those
>>> forms.
>>>     
>>
>>
>> For those keeping score, these are sequences A033220, A107135, A107136,
>> A107137.
>>
>> Tony
>>
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>>   
>
> __________ NOD32 Informacje 2701 (20071204) __________
>
> Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
> http://www.nod32.com lub http://www.nod32.pl
>
>