Raha ayaa ku boodaya khadka lambarka waxaadna isku dayeysaa inaad qabsato. Boodka iyo qabashada marwalba way isbedelayaan. Raha wuxuu ka bilaabmaa meesha \(s \in \mathbb{Z}\) dhaqaaq kasta oo ay sameysana wuxuu ku boodaa masaafo \(z \in \mathbb{Z}\) (haddii \(z>0\) , wuu boodayaa midigta, haddii kale bidix). \(z\) waa la mid boodbooyin kasta. Snapping wuxuu ka kooban yahay qeexida booska integer. Midna ma garanayo \(z\) iyo \(s\) midna. Waxaan muujineynaa inay jirto wado markasta lagu soo qabto raha.
Marka hore, \(a_1 = s\) iyo \(a_{n+1} = a_n + z = s + n \cdot z\) wata \(s,z \in \mathbb{Z}\) .
Waxaan dooranay hadda
$$h:\mathbb{N} \to \mathbb{Z}^2: h(2^k r) = \left ( (-1)^{k+1} \left \lfloor \frac{k+1}{2} \right \rfloor, (-1)^{\frac{r+1}{2}} \left \lfloor \frac{r+1}{4} \right \rfloor \right ) $$
sida shaqada u xilsaaraysa (si sax ah) tiro lambar ah tirooyin dhan lambar kasta oo dabiici ah. Xulashada shaqadan waa iyada oo loo marayo shaqooyinka \(f(n) = (-1)^n \left \lfloor \frac{n}{2} \right \rfloor\) , the \(\mathbb{N}\) saaran \(\mathbb{Z}\) iyo \(g(2^kr) = (k+1, \frac{r+1}{2})\) , oo \(\mathbb{N}\) saaran \(\mathbb{N}^2\) khariidad si asluubeysan, dhiirigalin leh.
Waxaan hadda muujineynaa qiyaasta \(h\) ( \(h\) sidoo kale waa cirbad, laakiin uma baahnin hantidan).
Ha \((x,y) = (2^{k_1} r_1, 2^{k_2} r_2) \in\mathbb{Z}^2\) . Laakiin markaa
$$h \left ( 2^{2 \cdot 2^{k_1} r_1 - 1} \cdot (4 \cdot 2^{k_2} r_2 - 1) \right ) = (2^{k_1} r_1, 2^{k_2} r_2) = (x,y).$$
Sidaa awgeed: \(\forall (s,z) \in \mathbb{Z}^2 \, \exists \, m \in \mathbb{N}\) leh \(h(m) = (x_m,y_m) = (s, z)\) .
Tusaale ahaan, haddii ay tahay waqtigeenna inaan u dhaqaaqno \(n = 88\) , waxaan xisaabineynaa \(h(88)=(2,3)\) waxaan doorannaa \(2 + 88 \cdot 3 = 266\) oo ah booska.
Ka dib marka sida saxda ah \(m\) ula dhaqaaqdo \(x_m + m \cdot y_m = s + m \cdot z = a_m\) doorashadu waxay ku \(x_m + m \cdot y_m = s + m \cdot z = a_m\) raha.
Marka lagu daro \(h\) , shaqooyin kale oo badan sida shaqada isku dheelitirka ee Cantor ama wareegga biiyaha ayaa suuragal ah.
Halkan waa hirgelinta fudud ee JavaScript:
See the Pen catch the frog by David Vielhuber (@vielhuber) on CodePen.