Question

now we see under what conditions,$(a+bI)^N=a-bI\pmod q$, we can conclude N dosent have $4k+1$ factors, suppose there is a factor $q=4k+1$ for $N$and$N=q*d$ , then $(a+bI)^N=((a+bI)^q)^d=(a-bI)^d=a^d+b^d(-I)^{4k+3}=a^d+b^d*I\pmod q$. so N dosent have $4k+1$ factors. and There are 3,or 5 or 7 etc.factors of form 4k+3 for N. now to the 3 factors. Suppose N exists. $(a+bI)^{q_1q_2q_3}=a-bI\pmod {q_1q_2q_3}$
$((a+bI)^{q_1})^{q_2q_3}=((a-bI)^{q_2q_3}=a-bI\pmod {q_1}$
$((a+bI)^{q_2})^{q_1q_3}=((a-bI)^{q_1q_3}=a-bI\pmod {q_2}$
$((a+bI)^{q_3})^{q_1q_2}=((a-bI)^{q_1q_2}=a-bI\pmod {q_3}$
as$((a-bI)^{(q_1+1)})^{(q_1-1)k_1}=(a^2+b^2)^{(q_1-1)k_1}=1\pmod {q_1}$
multiply (a-bI) to the both sides.
$(a-bI)((a-bI)^{(q_1+1)})^{(q_1-1)*k_1}=(a-bI)^{(q_1^2-1)*k_1+1}\pmod {q_1}$
$(a-bI)((a-bI)^{(q_2+1)})^{(q_2-1)*k_2}=(a-bI)^{(q_2^2-1)*k_2+1}\pmod {q_2}$
$(a-bI)((a-bI)^{(q_3+1)})^{(q_3-1)*k_3}=(a-bI)^{(q_3^2-1)*k_3+1}\pmod {q_3}$
so we have
$q_2q_3=(q_1^2-1)*k_1+1$ (1)
$q_1q_3=(q_2^2-1)*k_2+1$ (2)
$q_1q_2=(q_3^2-1)*k_3+1$ (3)
(1)/(2)
$\frac{q_2}{q_1}=\frac{(q_1^2-1)*k_1+1}{(q_2^2-1)*k_2+1}$
introduce variable $t$ and it must be a integer.
$(q_1^2-1)k_1+1=q_2t$ (4)
$(q_2^2-1)k_2+1=q_1t$ (5)
tanspose
$(q_1^2-1)k_1=q_2t-1$ (6)
$(q_2^2-1)k_2=q_1t-1$ (7)
(6)*(7)
$q_1q_2t^2-(q_1+q_2)t+1-(q_1^2-1)(q_2^2-1)k_1k_2=0$
$\Delta=(q_1+q_2)^2-4q_1q_2(1-(q_1^2-1)(q_2^2-1)k_1k_2)$
$=(q_1-q_2)^2+4q_1q_2(q_1^2-1)(q_2^2-1)k_1k_2$
$t=\frac{(q_1+q_2)\pm\sqrt{(q_1-q_2)^2+4q_1q_2(q_1^2-1)(q_2^2-1)k_1k_2}}{2q_1q_2}$
$(q_1+q_2)\pm\sqrt{\Delta}=(q_1+q_2)\pm(q_1-q_2)\ne 0\pmod {q_1q_2}$
so $t$ does not exist. therefore the Compsite Number for $N=4k+3$ dose not exist.