±q (aa+bb)(cc+dd)=(ac+bd)^2+(ad-bc)^2
and (4k+1)-type prime pj= pp+qq ¤èªk°ß¤@
(1)
when k=p
j1*X, which X has no any factor of p
j1
we know
k=uu+vv ¦pªG¦³w²Õ¤¬½è¸Ñ,«h, kk=ss+tt ¤]¦³w²Õ¤¬½è¸Ñ
; (¦pªG¦³«D¤¬½è¸Ñ,«h´£¥X¤½¦]¼Æ,¦A¥¤è)
XX=xx+yy ¦pªG¦³h²Õ¤¬½è¸Ñ,«h, X=ss+tt ¤]¦³h²Õ¤¬½è¸Ñ (¦pªG¦³«D¤¬½è¸Ñ,«h´£¥X¤½¦]¼Æ,¦A¥¤è)
kk=pj1*pJ1*XX=pj1*pj1*(xx+yy)
=(pp+qq)(pp+qq)(xx+yy)
=(pp+qq)((px+qy)^2+(py-qx)^2)
=(pp+qq)((px-qy)^2+(py+qx)^2)
=(ppx+2pqy-qqx)^2+(ppy-2pqx+qqx)^2
=(ppx-2pqy-qqx)^2+(ppy+2pqx+qqx)^2
=(ppx+qqx)^2 +(ppy+qqy)^2
==> if XX=xx+yy ¦pªG¦³h²Õ¸Ñ, then kk=ss+tt ¦³3*h²Õ¸Ñ
==> if X=xx+yy ¦pªG¦³h²Õ¸Ñ, then k=ss+tt ¦³3*h²Õ¸Ñ
(2)
when k=p
j1^2*X, which X has no any factor of p
j1
let X=xx+yy ¦³h²Õ¸Ñ
k=pj1*pJ1*X=pj1*pj1*(xx+yy)
=(pp+qq)(pp+qq)(xx+yy)
=(pp+qq)((px+qy)^2+(py-qx)^2)
=(pp+qq)((px-qy)^2+(py+qx)^2)
=(ppx+2pqy-qqx)^2+(ppy-2pqx+qqx)^2
=(ppx-2pqy-qqx)^2+(ppy+2pqx+qqx)^2
=(ppx+qqx)^2 +(ppy+qqy)^2
==> k=ss+tt ¦³3*h²Õ¸Ñ
==> if X=xx+yy ¦pªG¦³h²Õ¸Ñ, then k=ss+tt ¦³3*h²Õ¸Ñ
(3)
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µL¿ú¤u
ªº¦P²z,¦Û¤v·QºO
)
when k=p
j1^t1*X, which X has no any factor of p
j1
if X=xx+yy ¦³h²Õ¸Ñ
then k=ss+tt ¦³(1+2*[(t1+1)/2])*h²Õ¸Ñ
(4) when k=1, N(k)=4
(5) when k=(pi1*pi2*...*Pin)^2,(Pi are (4k-1)-type primes), N(k)=4
(6) when k=2^(2n), N(k)=4
when k=2^(3n+1), N(k)=4
(7) from (3)(4)(5)(6), we get:
If k=2^c* (pi1*pi2*pi3*...*pim)^2*(pj1^t1)*(pj2^t2)*....*(pjn^tn),
pi:(4k-1)-type primes
pj:(4k+1)-type different primes
then,
N(k)= 4* (1+2*[(t1+1)/2] )*(1+2*[(t2+1)/2] )*...*(1+2*[(tn+1)/2] )
---
else: N(0)=1
else: N(k)=0