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)格點數(shù)

g(r) 半徑為r的圓上格點數(shù)

x格點的橫坐標(biāo)

y格點的縱坐標(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、半徑為r的圓的內(nèi)部有多少個格點？

三、格點的分類

原點：x=y=0.

分點：x2=y2≠0.

正點：x×y=0，x2≠y2.

散點：x×y≠0，x2≠y2.

四、圓上格點

x2+y2=r2.

∵x∈Z，∴x2∈Z，

∵y∈Z，∴y2∈Z，

∴x2+y2=r2∈Z.

∴當(dāng)r2?Z時，g(r)=0.

∵g(0)=1，

∴只考慮r2∈Z，r>0.

正點：當(dāng)r∈Z時，有4個.

五、圓內(nèi)格點

x2+y2≤r2.

∵當(dāng)r2?Z時，g(r)=0,

∴圓內(nèi)格點是分層分布的，它們只在半徑為r且r2∈Z的同心圓上.

原點：1.

分點：4s.

正點：4[r].

六、計算器草圖

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.