Homework SourseProve the following NOTE we are referring to congruent numbe
Prove the following:
NOTE: we are referring to congruent numbers as: A positive integer N is called a congruent number when there is a rational right triangle with area N.
Prove that if N is a congruent number and m is any positive integer, then Nm2 is also a congruent number. Also prove that if N is a congruent number, m is a positive integer and m2 divides N, then N / m2 is also a congruent number.
Solution
We know that if a positive rational number N is a congruent number, then there exists a rational right triangle with area N i.e. there exist rational numbers p, q, r > 0 such that p2 + q2 = r2 and (1/2)pq = N. Now, if m is a positive integer, then the numbers mp, mq, mr > 0 are also rational numbers . Also, (mp)2 + (mq)2 = m2p2 + m2 q2 = m2 r2 = (mr)2. Further, (1/2)(mp)(mq) = N*m*m = Nm2. This means that there exists a rational right triangle with area Nm2. Also, Nm2 is a positive rational number as N is a positive rational number and m2 is a positive integer. Hence, Nm2 is also a congruent number.
If N is a congruent number, m is a positive integer and m2 divides N, then N being a positive rational number, N/m2 is also a positive rational number ( as m2 is also a positive integer). Now, since N is a congruent number, then there exist rational numbers p, q, r > 0 such that p2 + q2 = r2 and (1/2)pq = N. Further, since m is a positive integer, therefore, p/m , q/m and r/m are also positive rational numbers. Also, (p/m)2+(q/m)2 = (p2/m2)+( q2/m2) = (p2 + q2)/m2= r2/m2 = (r/m)2. Also, (1/2)(p/m)(q/m) = N/(m*m) = N/m2. Thus, there exists a rational right triangle with area N/m2. Hence, N /m2 is also a congruent number.
