[seqfan] Re: defn of A167415

[israel at math.ubc.ca]

This is very strange. You should have solution y = -x*A^2*(A^2+3*x)^(-1) 
(mod n) whenever A^2 + 3*x is coprime to n. The only cases where no nonzero 
x will work are when A and n are both divisible by 3, and n=2 with A odd.
   

On Jun 14 2019, Richard J. Mathar wrote:

> How is A167415 defined? It constructs solutions of A^2*(x+y)+3*x*y=0 (mod 
> n) and lists n such only the primitive solution (x=0,y=0) exists. 
> Presumably A may be any non-zero (effectively: positive) integer (?).
>
> Construct for example n=3, A=1, x=1, y=2 with 1^2*(1+2)+3*1*2 = 9 ==0 
> (mod 3) and with that kind of definition, n=3 should not be in the 
> sequence.
>
> Construct for example n=6, A=1, x=2, y=4 with 1^2*(2+4)+3*2*4 = 30 ==0 
> (mod 6) and with that kind of definition, n=6 should not be in the 
> sequence.
>
>I considered the idea that only nontrivial coprime (x,y) are allowed
>but that does not resolve the case of n=3 as shown above , and also
>not n=6 because
>"n=", 6, "A=", 4, "x=", 1, "y=", 2, with 4^2*(1+2)+3*1*2=54 == 0 (mod 6)
>works as counterexample.
>
> Further restriction to coprime (A,x,y) does not work either. See "n=", 
> 13, "A=", 1, "x=", 1, "y=", 3, with 1^2*(1+3)+3*1*3 = 13 == 0 (mod 13).
>
>Any ideas to elicidate the definition?
>
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>
>