N/2+1=P N/4+1=Q (P, Q, odd squares)
N=4(P-Q)=4(Q-1) [so: P-Q=Q-1].
P-Q is a difference of odd squares which is always congruent to 0 (mod 8).
Q-1=q^2-1=(q+1)(q-1). 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 Q-1 is congruent to 0 (mod 3), then both (P-Q) and (Q-1) should be congruent with 0 (mod 24) (as P-Q=Q-1). Then N=4(P-Q) should be congruent to 0 (mod 96).
b) If Q-1 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(2n-1))^2
P is also an odd square. Then P=(2m-1)^2
P-Q=(2m-1)^2-(6n-3)^2=4(m(m-1)-9n(n-1)-2)
This expression is always congruent to 0 or 1 (mod 3). But Q-1 is congruent to 2 (mod 3). So it can't be P-Q=Q-1.Then no possible value for N.
Edited on June 7, 2016, 4:32 pm
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Posted by armando
on 2016-06-06 08:40:40 |
Another way
