Re: Question on ternary quadratic form

[hermann@stamm-wilbrandt.de]

On 2023-11-20 23:18, Bill Allombert wrote:
> The difference is that tqf is calling ispseudoprime on numbers of size 
> n, while
> threesquares only call ispseudoprime on numbers of size about sqrt(n).
>
> If you like to have an assert() that you can easily disable, you can
> define it as
>
> assert(b,s="assert") = if(!b(),error(Str(s)));
>
> and then replace
>
> assert(COND)
> by
> assert(()->COND)
>
> This way you can disable assert() by refinining it as
> assert(b,s)=;
> and COND will not be evaluated.
Thanks for the assert trick!

And for the ispseudoprime() explanation n vs sqrt(n).


The initial ternary quadratic form matrix generation goes back to 
Dirichlet 1850.
I did brute force determine all matrices

? Q

[a b 1]

[b c 0]

[1 0 n]

?

with matdet(Q)==1 ( and qfeval(Q,[0,0,1]~)==n ) for a,b,c up to some 
factor of n:
https://www.mersenneforum.org/showthread.php?p=643698#post643698

Interesting to see other ternary quadratic forms result in different 
sums of squares:

? {Qs=[
 [2,16,1;16,129,0;1,0,65]
,[26,71,1;71,194,0;1,0,65]
,[27,59,1;59,129,0;1,0,65]
,[30,108,1;108,389,0;1,0,65]
,[38,70,1;70,129,0;1,0,65]
,[78,123,1;123,194,0;1,0,65]
,[99,113,1;113,129,0;1,0,65]
,[163,145,1;145,129,0;1,0,65]
,[203,281,1;281,389,0;1,0,65]
,[274,188,1;188,129,0;1,0,65]
,[307,199,1;199,129,0;1,0,65]
,[362,265,1;265,194,0;1,0,65]
];}
? foreach(Qs,Q,print((qflllgram(Q)^-1)*[0,0,1]~))
[-8, 0, 1]~
[2, -5, -6]~
[-2, -5, 6]~
[-7, 0, -4]~
[-2, -5, -6]~
[5, 2, -6]~
[8, 0, -1]~
[-8, 0, -1]~
[-7, 0, 4]~
[-2, -5, -6]~
[-2, -5, 6]~
[2, -5, 6]~
?

Would be nice to identify how to get those containing 0^2 ;-)
Such "sum of two squares" would allow to factor n easily:

? gcd(65, (8+4)^2 + (1-7)^2)
%49 = 5
? gcd(65, (8+4)^2 + (1+7)^2)
%50 = 13
?


Regards,

Hermann.