Abstract
An algorithm to find a number T given a particular binary quadratic form.
Given a number T the question is whether we can write in a binary quadratic form.
\(Let\ T=Ax^2+Bxy+Cy^2\) then ; How can we write T in terms of the coefficients A, B and C if we specify that all the letters involved must be integers T,A, B,C, x,y \(T,A,B,C,x,y\ \ belong\ in\ Z\)
Let us first give an example. Suppose we have 17 as the number we want to write down. We can write it as \(17=x^2+y^2=4^2+1^2\); and if we want to write the number 43, we can write it thus; \(43=6^2+6.1+1^2=x^2+xy+y^2\).
Then the question of representation arises; how can we write any number as a binary quadratic form? Given that \(Ax^2+Bxy+Cy^2-T=0\), find general expressions of numbers A, B, C, T such that x and y are integers. In other words given an integer T find the way it can be represented by A, B, C.
Proposition:
If \(A=\left(PM-pm\right)\) \(B=\left\{\left(PN+QM\right)-\left(pn+qm\right)\right\}\) \(C=\left(QN-qn\right)\), then
\(T=\frac{\left(M+N-m-n\right)c_1+\left(P+Q-p+q\right)}{m+n}\) with \(m+n=1;\) \(Pq-pQ=1\)
Proof:
Let us write \(\left(Ax^2+Bxy+Cy^2-T\right)=GH-JK=0\ \ \ \ \ \ \ \ \ \left(1\right)\)
where \(G=\left(Px+Qy+c_1\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(2\right)\ \)
and \(H=\left(Mx+Ny+c_2\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3\right)\)
and \(J=\left(px+qy+c_3\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(4\right)\)
and \(K=\left(mx+ny+c_4\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(5\right)\)
Now we expand the expression \(GH-JK=0\ \ \ or\ \ \left(2\right)\left(3\right)-\left(4\right)\left(5\right)=0\) in the above equations.
\(\left(Px+Qy+c_1\right)\left(Mx+Ny+c_2\right)-\left(px+qy+c_3\right)\left(mx+ny+c_4\right)=0\ \ \left(6\ \ \right)\)
Expanding the above expression (6) and gathering all terms as powers of x and y;
\(\left(PM-pm\right)x^2=Ax^2\ \ \ \ \ \ \ \ \ \ \ \ \left(7\right)\)
\(\left\{\left(PN+QM\right)-\left(pm-qn\right)\right\}xy=Bxy\ \ \ \ \ \ \ \left(8\right)\)
\(\left(QN-qn\right)y^2=Cy^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(9\right)\)
\(\left\{\left(Qc_2+Nc_1\right)-\left(qc_4+nc_3\right)\right\}y=Dy=0y\ \ \ \ \ \ \left(10\right)\)
\(\left\{\left(Pc_2+Mc_1\right)-\left(pc_4+mc_3\right)\right\}x=Ex=0x\ \ \ \ \ \left(11\right)\)
\(c_1c_3-c_2c_4=-T\ \ \ \ \ \ \ \ \ \ \left(12\right)\)
We want to introduce conditions for integer solutions;
We say that \(Pq-pQ=+1\)
This is derived from the two equations ;
\(Px+Qy+c_1=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(14\right)\)
\(px+qy+c_3=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(15\right)\)
\(GH-JK=0H-0K=0\ \ \ \ \ \ \ \ \ \ \left(16\right)\)
Notice that the choice of (14) and (15) make the expression zero.
\(c_2-c_1=1,\ \ \ \ \ therefore\ \ c_3-c_1=-T\)
We add equation (10) to (11) and simplify;
\(\left\{\left(\left[P+Q\right]c_2+\left[M+N\right]c_1-\left(\left[p+q\right]c_4+\left[m+n\right]c_3\right)\right)\right\}=D+E=0\)
Replacing c3, i.e. \(c_3=c_1-T\), we have:
\(\left[\left\{\left(P+Q\right)-\left(p+q\right)\right\}+\left\{\left(M+N\right)c_1-\left(m+n\right)\left(c_1-T\right)\right\}\right]=0\)
Solving for T in the above equation we have; \(T=\frac{\left(M+N-m-n\right)c_1+\left(P+Q-p-q\right)}{m+n}\).
Notice that \(c_1\) is a free variable free to take on any integer value . So the above expression is an expression of any integer T.
Algorithm
This suggest an algorithm for finding a number T given a particular binary quadratic form.
1 Write down the binary quadratic equation as the difference of two products GH-JK=0.
2. Set (m+n)=1. This is to ensure that T is an integer rather than a fraction.
3. Set (Pq-pQ)=1. This will ensure that the solution (x,y) is made up of integers x and y.
4. Set [(M+N)-(m+n)] a small integer (but not always).
5. Then find \(c_1\) from the equation \(T=c_1Q_1+R\)
6. Then use P,Q,M,N, m,n,p,q, to find A, B, C.
