Primes Produced by Quadratic Forms

[T. D. Noe]

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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