Triplets Pythagoriciens

(a, b, c)
a2+b2=c2.(∗)
a4+b4=c4.
a b c
an+bn=cn,
n2
n= 3 n= 5
b= 0 c= 0

(∗)c2
cos θ=1−t2
1 + t2,
sin θ=2t
1 + t2,
t= tan(θ/2)
θ t t =p
q
a2+b2=c2.
(3,4,5) 32+ 42= 9 + 16 = 25 = 52
b= 0 c= 0 a=b= 0
c2(∗)
α2+β2= 1,
α=a/c β =b/c
cos2θ+ sin2θ= 1.
t
p/q θ = 2Arctan (p/q) Arctan tan
tan(Arctan (x)) = x t =p/q
1−p2
q22
1 + p2
q22+2p
q2
1 + p2
q22= 1,
1−p2
q22
+2p
q2
=1 + p2
q22
.
q4
q2−p22+ (2pq)2=q2+p22.
p= 2 q= 3
q2−p2= 5
2pq = 12
q2+p2= 13.

52+ 122= 25 + 144 = 169 = 132
1
a, b c a b c2=ab
a b a0b0a=a02b=b02
a b
a b a b
c2=c2
1c2 2· · · c2 nn∈Nc1, c2,· · · , cn
a b c
kakbkc26p1< p2,· · · < pka26q1< q2<· · · < qkb26r16r26· · · 6rkc
n1, n2,· · · , nk>1m1, m2,· · · , mk>1s1, s2,· · · , sk>1
a=pn1
1pn2 2· · · pnk
k, b =qm1
1qm2 2· · · qmkb
k, c =rs1
1rs2 2· · · rskc
kc.
i, j pi6=qja b ni>1pia
b j qjb mj>1pi6=qj
c2=r2s1
1r2s2 2· · · r2skc
kc=ab =pn1
1pn2 2· · · pnk
k×qm1
1qm2 2· · · qml
k.
c2pi6=qjni
mjni= 2n0
imj= 2m0 j
a=pn0
1
1pn0
2
2· · · pn0
k
k2
, b =qm0
1
1qm0
2
2· · · qm0
kb
k2
,
a b
a2=u2−v2, b2= 2uv, c =u2+v2.
a, b, c
a2+b2=c2.
a=u2−v2, b = 2uv, c =u2+v2,
u v a2+b2=c2
a b c
a b
c−a c +a2
u v
c−a= 2u2, c +a= 2v2.
(a, b, c)
a2+b2=c2

4k+ 1
(c−a)(c+a) = c2−a2=b2
a b a2+b2=c2a, b, c
u v
a=v2−u2, b = 2uv, c =u2+v2.
a b a = 2k+ 1, b = 2k0+ 1
a2+b2= (2k+ 1)2+ (2k0+ 1)2= 4k2+ 4k+ 1 + 4k02+ 4k0+ 1 = 4(k2+k+k02+k0) + 2 = c2.
c2c c = 2c0
4c02= 4(k2+k+k02+k0) + 2,2 = 4(c02−(k2+k+k02+k0))
4 2
a b a2b2a2+b2=c2c2c
a b c
a b
c
d c −a c +a d 2c2a
c a d 2c a
2c−a c +a c −a c +a
2
(c−a)(c+a) = c2−a2=b2
c−a= 2x, c +a= 2y, b = 2b0,
x y
xy = (b0)2.
x y
c−a= 2u2, c +a= 2v2.
c=u2+v2.
a=v2−u2.
b2= 4u2v2,
b= 2uv
(a, b, c)a2+b2=c2
a2+b2=c2
a2= (v2−u2)2=v4−2u2v2+u4, c2= (v2+u2)2=v4+ 2u2v2+u4.

a4+b4=c4.
a, b, c
(a, b, c)
a4+b4=c2.
c2c4
a b c
a b
u v u v
a2=u2−v2, b2= 2uv, c =u2+v2.
x, y, z, t z t
u=x2, v = 2y2, a2=z2−t2, v = 2zt, u =z2+t2.
z t z =z2
1t=t2 1
x2=z4
1+t4 1.
x<c
a4+b4=c4
a, b, c a4+b4=c4a, b, c
a=da0, b =db0, c =dc0, a0, b0, c0.
d4a04+b04=c04
(a2, b2, c)a2
b2a b c
u v
a2=u2−v2, b2= 2uv, c =u2+v2.