Lele a puni kahi rana ma kahi laina helu, a ʻo ka pahuhopu ka hopu ʻana iā ia. Lele a hopu ʻokoʻa. Hoʻomaka ka rana ma ke kūlana \(s \in \mathbb{Z}\) a me kēlā me kēia neʻe ʻana, lele ia i kahi mamao o \(z \in \mathbb{Z}\) (inā \(z>0\) , lele ia i ka ʻākau; inā ʻaʻole, i ka hema). Ua like ʻo \(z\) no kēlā me kēia lele. ʻO ka hopu ʻana i ka rana e pili ana i ka wehewehe ʻana i kahi kūlana helu piha. ʻAʻole ʻike ʻia \(z\) a me \(s\) . E hōʻike mākou aia kahi kaʻina hana e hiki ai iā mākou ke hopu i ka rana.
ʻO ka mea mua, \(a_1 = s\) a me \(a_{n+1} = a_n + z = s + n \cdot z\) me \(s,z \in \mathbb{Z}\) .
Koho mākou i kēia manawa
$$h:\mathbb{N} \to \mathbb{Z}^2: h(2^k r) = \left ( (-1)^{k+1} \left \lfloor \frac{k+1}{2} \right \rfloor, (-1)^{\frac{r+1}{2}} \left \lfloor \frac{r+1}{4} \right \rfloor \right ) $$
e like me ka hana e hāʻawi (pololei) i kahi tuple helu o nā helu holoʻokoʻa i kēlā me kēia helu kūlohelohe. ʻO ke koho o kēia hana ma o nā hana \(f(n) = (-1)^n \left \lfloor \frac{n}{2} \right \rfloor\) , ka \(\mathbb{N}\) ma \(\mathbb{Z}\) a me \(g(2^kr) = (k+1, \frac{r+1}{2})\) , ʻo wai \(\mathbb{N}\) ma \(\mathbb{N}^2\) palapala ʻāina bijectively, hoʻoikaika.
Hōʻike mākou i kēia manawa i ka surjectivity o \(h\) ( \(h\) mea hoʻopili pū, akā ʻaʻole pono mākou i kēia waiwai).
E ʻae \((x,y) = (2^{k_1} r_1, 2^{k_2} r_2) \in\mathbb{Z}^2\) . Akā laila
$$h \left ( 2^{2 \cdot 2^{k_1} r_1 - 1} \cdot (4 \cdot 2^{k_2} r_2 - 1) \right ) = (2^{k_1} r_1, 2^{k_2} r_2) = (x,y).$$
No laila: \(\forall (s,z) \in \mathbb{Z}^2 \, \exists \, m \in \mathbb{N}\) me \(h(m) = (x_m,y_m) = (s, z)\) .
ʻO kahi laʻana, inā ʻo kā mākou neʻe e neʻe ma \(n = 88\) , helu mākou \(h(88)=(2,3)\) a koho iā \(2 + 88 \cdot 3 = 266\) ma ke ʻano he kūlana.
A ma hope o ka neʻe pololei ʻana o \(m\) me \(x_m + m \cdot y_m = s + m \cdot z = a_m\) hāʻule ke koho ma ka rana.
Ma waho o \(h\) , hiki i nā hana ʻē aʻe he nui e like me ka hana pairing a Cantor a i ʻole kahi ʻaoʻao bijective .
Eia kahi hana maʻalahi ma JavaScript:
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