1. a perfect square

Let n=pq and q-p=2d where d>0 and p, q are odd prime numbers.
1-Show that n+d2 is a perfect square i.e. n+d2=m2 for some integer m.
2-If you know that n+d2 is a perfect square, can you find prime factor of n
how can I show that, any help will be apprciated.

2. Originally Posted by koko2009
Let n=pq and q-p=2d where d>0 and p, q are odd prime numbers.

1-Show that n+d2 is a perfect square i.e. n+d2=m2 for some integer m.
2-If you know that n+d2 is a perfect square, can you find prime factor of n

how can I show that, any help will be apprciated.

Plain substitution: $n + d^2 = pq+\left(\frac{q-p}{2}\right)^2=
pq+\frac{q^2}{4}-\frac{qp}{2}+\frac{p^2}{4}$

Well, now just check the above is a perfect square (some junior high school algebra is required here)

Tonio

3. Originally Posted by tonio
Plain substitution: $n + d^2 = pq+\left(\frac{q-p}{2}\right)^2=
pq+\frac{q^2}{4}-\frac{qp}{2}+\frac{p^2}{4}$

Well, now just check the above is a perfect square (some junior high school algebra is required here)

Tonio
$thanks, but how about saying that square (p+q)^2/2 is n+d^2 so n+d^2 is perfect sequare. also, how can we find the prime factor of n$

4. Originally Posted by koko2009
Let n=pq and q-p=2d where d>0 and p, q are odd prime numbers.

1-Show that n+d2 is a perfect square i.e. n+d2=m2 for some integer m.
2-If you know that n+d2 is a perfect square, can you find prime factor of n

how can I show that, any help will be apprciated.
(2) $n+d^2 = m^2$

$n = m^2-d^2$

$n = (m+d)(m-d)$

q=(m+d) & p=(m-d)

or: p=(m+d) & q=(m-d)

.

5. Originally Posted by koko2009
$thanks, but how about saying that square (p+q)^2/2 is n+d^2 so n+d^2 is perfect sequare. also, how can we find the prime factor of n$
"How about saying that..."? You have to prove it, of course. And abou that odd question about THE odd prime factor of n: n = qp, so both q, p are odd factors of n...

Tonio