N/2+1=P N/4+1=Q (P, Q, odd squares)
N=4(PQ)=4(Q1) [so: PQ=Q1].
PQ is a difference of odd squares which is always congruent to 0 (mod 8).
Q1=q^21=(q+1)(q1). This is congruent:
 to 0 (mod 3) if q is not multiple of 3.
 to 2 (mod 3) if q is multiple of 3.
a) If Q1 is congruent to 0 (mod 3), then both (PQ) and (Q1) should be congruent with 0 (mod 24) (as PQ=Q1). Then N=4(PQ) should be congruent to 0 (mod 96).
b) If Q1 is congruent to 2 (mod 3) this is because Q=q^2 is and odd square multiple of 9 (q is multiple of 3: an odd one of course). Then it can be expressed as: Q=(3(2n1))^2
P is also an odd square. Then P=(2m1)^2
PQ=(2m1)^2(6n3)^2=4(m(m1)9n(n1)2)
This expression is always congruent to 0 or 1 (mod 3). But Q1 is congruent to 2 (mod 3). So it can't be PQ=Q1.Then no possible value for N.
Edited on June 7, 2016, 4:32 pm

Posted by armando
on 20160606 08:40:40 