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      格點(diǎn)計(jì)算器

      2020-03-31 10:59:36張奎福
      數(shù)理化解題研究 2020年9期
      關(guān)鍵詞:正點(diǎn)分點(diǎn)松原市

      張奎福

      (吉林省松原市長嶺縣巨寶山鎮(zhèn) 131533)

      一、符號

      [a] 不大于a的最大整數(shù)

      G(r) 半徑為r的圓內(nèi)格點(diǎn)數(shù)

      g(r) 半徑為r的圓上格點(diǎn)數(shù)

      x格點(diǎn)的橫坐標(biāo)

      y格點(diǎn)的縱坐標(biāo)

      r∈Zr是整數(shù)

      r?Zr不是整數(shù)

      ∑ 求和

      二、問題

      Gauss(1777-1855)問題:中心在原點(diǎn)、半徑為r的圓的內(nèi)部有多少個(gè)格點(diǎn)?

      三、格點(diǎn)的分類

      原點(diǎn):x=y=0.

      分點(diǎn):x2=y2≠0.

      正點(diǎn):x×y=0,x2≠y2.

      散點(diǎn):x×y≠0,x2≠y2.

      四、圓上格點(diǎn)

      x2+y2=r2.

      ∵x∈Z,∴x2∈Z,

      ∵y∈Z,∴y2∈Z,

      ∴x2+y2=r2∈Z.

      ∴當(dāng)r2?Z時(shí),g(r)=0.

      ∵g(0)=1,

      ∴只考慮r2∈Z,r>0.

      正點(diǎn):當(dāng)r∈Z時(shí),有4個(gè).

      五、圓內(nèi)格點(diǎn)

      x2+y2≤r2.

      ∵當(dāng)r2?Z時(shí),g(r)=0,

      ∴圓內(nèi)格點(diǎn)是分層分布的,它們只在半徑為r且r2∈Z的同心圓上.

      原點(diǎn):1.

      分點(diǎn):4s.

      正點(diǎn):4[r].

      六、計(jì)算器草圖

      七、VB編程

      Dim a, b, c, r, s, t, i, l, k

      Private Sub CommandButton1_Click()

      a = 0: b = 0: c = 1: r = 0: s = 0: t = 0: i = 0: k = 0: TextBox1 = ""

      End Sub

      Private Sub CommandButton10_Click()

      If TextBox1 = 0 Then TextBox1 = "7": GoTo 10

      TextBox1 = TextBox1 & "7"

      10 End Sub

      Private Sub CommandButton11_Click()

      If TextBox1 = 0 Then TextBox1 = "8": GoTo 10

      TextBox1 = TextBox1 & "8"

      10 End Sub

      Private Sub CommandButton12_Click()

      If TextBox1 = 0 Then TextBox1 = "9": GoTo 10

      TextBox1 = TextBox1 & "9"

      10 End Sub

      Private Sub CommandButton13_Click()

      If Int(r) = r Then b = b + 4

      If Sqr(a / 2) = s Then b = b + 4

      For i = s + 1 To Sqr(a - 1)

      k = a - i * i

      If Int(Sqr(k)) = Sqr(k) Then b = b + 8

      Next i

      TextBox1 = b

      End Sub

      Private Sub CommandButton14_Click()

      If TextBox1 <> 0 Then TextBox1 = TextBox1 & "0"

      End Sub

      Private Sub CommandButton15_Click()

      For i = 1 To s:

      c = c + 8 * Int(Sqr(a - i * i))

      Next i

      c = c + 4 * Int(r) - 4 * s * s

      TextBox1 = c

      End Sub

      Private Sub CommandButton2_Click()

      If t = 0 Then TextBox1 = TextBox1 & ".": t = 1

      End Sub

      Private Sub CommandButton3_Click()

      a = TextBox1: r = Sqr(a): s = Int(Sqr(a / 2))

      TextBox1 = r

      End Sub

      Private Sub CommandButton4_Click()

      If TextBox1 = 0 Then TextBox1 = "1": GoTo 10

      TextBox1 = TextBox1 & "1"

      10 End Sub

      Private Sub CommandButton5_Click()

      If TextBox1 = 0 Then TextBox1 = "2": GoTo 10

      TextBox1 = TextBox1 & "2"

      10 End Sub

      Private Sub CommandButton6_Click()

      If TextBox1 = 0 Then TextBox1 = "3": GoTo 10

      TextBox1 = TextBox1 & "3"

      10 End Sub

      Private Sub CommandButton7_Click()

      If TextBox1 = 0 Then TextBox1 = "4": GoTo 10

      TextBox1 = TextBox1 & "4"

      10 End Sub

      Private Sub CommandButton8_Click()

      If TextBox1 = 0 Then TextBox1 = "5": GoTo 10

      TextBox1 = TextBox1 & "5"

      10 End Sub

      Private Sub CommandButton9_Click()

      If TextBox1 = 0 Then TextBox1 = "6": GoTo 10

      TextBox1 = TextBox1 & "6"

      10 End Sub

      Private Sub UserForm_Click()

      End Sub

      八、使用說明

      九、程序測試

      r2,G(r).

      1,5.10,37.102,317.103,3149.104,31417.105,314197.

      106,3141549.107,31416025.108,314159053.109,3141592409.

      1010,31415925457.1011,314159264013.1012,3141592649625.

      1013,31415926532017.1014,314159265350589.

      2*1014,628318530711997.3*1014,942477796083985.

      318309886183721-318309886183724,1015-3.

      318309886183725,1015+45.

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