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A351986
Four-column table read by rows, giving quadruples of integers [w,x,y,z] such that y^2 - y - x*z = 0 and x^2 = w*y with w*y != 0 and y != 1, sorted by the absolute value of y with the negatives first, then by x in ascending order.
2
-4, -2, -1, -1, -1, -1, -1, -2, -1, 1, -1, 2, -4, 2, -1, 1, -18, -6, -2, -1, -2, -2, -2, -3, -2, 2, -2, 3, -18, 6, -2, 1, 2, -2, 2, -1, 2, 2, 2, 1, -48, -12, -3, -1, -12, -6, -3, -2, -3, -3, -3, -4, -3, 3, -3, 4, -12, 6, -3, 2, -48, 12, -3, 1, 12, -6, 3, -1, 3, -3, 3, -2, 3, 3, 3, 2, 12, 6, 3, 1
OFFSET
1,1
COMMENTS
When [w,x,y,z] is a row, f(a,b,c) = w*a*b*c + x*(a*b + a*c + b*c) + y*(a+b+c) + z is associative in the following sense. f((a,b,c),d,e) = f(a,f(b,c,d),e) = f(a,b,f(c,d,e)) for all a,b,c,d,e. f(a,b,c) is commutative because of its symmetry.
For each quadruple, the corresponding f(a,b,c) has a unique zero element (call it theta), meaning f(a,b,theta) = f(a,theta,b) = f(theta,a,b) = theta for all a,b. Theta = -y/x = - x/w. f(a,b,c) also has not one but two identity elements (id_1 and id_2), meaning f(a,id_1,id_1) = f(id_1,a,id_1) = f(id_1,id_1,a) = a for all a and f(a,id_2,id_2) = f(id_2,a,id_2) = f(id_2,id_2,a) = a for all a. Id = (-y +- sqrt(y))/x = theta +- sqrt(y)/x. Thus theta = (id_1 + id_2)/2.
The identity elements and theta are integers when y is a square and x divides sqrt(y).
EXAMPLE
Table begins:
[ w, x, y, z]
-------------------
[ -4, -2, -1, -1];
[ -1, -1, -1, -2];
[ -1, 1, -1, 2];
[ -4, 2, -1, 1];
[-18, -6, -2, -1];
[ -2, -2, -2, -3];
[ -2, 2, -2, 3];
[-18, 6, -2, 1];
[ 2, -2, 2, -1];
[ 2, 2, 2, 1];
[-48, -12, -3, -1];
[-12, -6, -3, -2];
[ -3, -3, -3, -4];
[ -3, 3, -3, 4];
[-12, 6, -3, 2];
[-48, 12, -3, 1];
[ 12, -6, 3, -1];
[ 3, -3, 3, -2];
[ 3, 3, 3, 2];
[ 12, 6, 3, 1];
...
PROG
(PARI) { my(y=1); fordiv (y^2+y, x, print([-((y^2+y)/x)^2/y, -(y^2+y)/x, -y, -x]) );
fordiv (y^2+y, x, print([-(x^2/y), x, -y, (y^2+y)/x]) );
for (y = 2, 6, fordiv (y^2+y, x, if(type(w = -(((y^2+y)/x)^2)/y)=="t_INT", print([w, -(y^2+y)/x, -y, -x]) ));
fordiv (y^2+y, x, if(type(w = -x^2/y)=="t_INT", print([w, x, -y, (y^2+y)/x]) ));
fordiv (y^2-y, x, if(type(w = (((y^2-y)/x)^2)/y)=="t_INT", print([w, -(y^2-y)/x, y, -x]) ));
fordiv (y^2-y, x, if(type(w = x^2/y)=="t_INT", print([w, x, y, (y^2-y)/x]) )) )}
CROSSREFS
Cf. A332083.
The rows of A351581 are a subset.
Sequence in context: A266144 A377366 A016508 * A325496 A010312 A303599
KEYWORD
sign,tabf
AUTHOR
David Lovler, Feb 27 2022
STATUS
approved