A sphenic number S.N. is a product of three distinct prime numbers.
a. Clearly each S.N. has 8 divisors. Show that not only S.N.s claim this feature.
b. Find the smallest consecutive pair n,n+1 of sphenic numbers.
c. Same for the smallest triplet.
d. Prove that sphenic quadruplet n,n+1,n+2,n+3 is "mission impossible".
a. 2^3 * 3 = 24 has this feature, as does any number that's a prime squared times another prime. The divisors of 24 are 1, 2, 4, 8, 3, 6, 12 and 24.
b. and c. All the pairs and triplets below n=3000 are tabulated below. The first pair is 230,231 and the first triplet is 1309-1311. There are four sets of triplets in the list:
230-231 2
230 = 2 * 5 * 23
231 = 3 * 7 * 11
285-286 2
285 = 3 * 5 * 19
286 = 2 * 11 * 13
429-430 2
429 = 3 * 11 * 13
430 = 2 * 5 * 43
434-435 2
434 = 2 * 7 * 31
435 = 3 * 5 * 29
609-610 2
609 = 3 * 7 * 29
610 = 2 * 5 * 61
645-646 2
645 = 3 * 5 * 43
646 = 2 * 17 * 19
741-742 2
741 = 3 * 13 * 19
742 = 2 * 7 * 53
805-806 2
805 = 5 * 7 * 23
806 = 2 * 13 * 31
902-903 2
902 = 2 * 11 * 41
903 = 3 * 7 * 43
969-970 2
969 = 3 * 17 * 19
970 = 2 * 5 * 97
986-987 2
986 = 2 * 17 * 29
987 = 3 * 7 * 47
1001-1002 2
1001 = 7 * 11 * 13
1002 = 2 * 3 * 167
1022-1023 2
1022 = 2 * 7 * 73
1023 = 3 * 11 * 31
1065-1066 2
1065 = 3 * 5 * 71
1066 = 2 * 13 * 41
1085-1086 2
1085 = 5 * 7 * 31
1086 = 2 * 3 * 181
1105-1106 2
1105 = 5 * 13 * 17
1106 = 2 * 7 * 79
1130-1131 2
1130 = 2 * 5 * 113
1131 = 3 * 13 * 29
1221-1222 2
1221 = 3 * 11 * 37
1222 = 2 * 13 * 47
1245-1246 2
1245 = 3 * 5 * 83
1246 = 2 * 7 * 89
1265-1266 2
1265 = 5 * 11 * 23
1266 = 2 * 3 * 211
1309-1311 3
1309 = 7 * 11 * 17
1310 = 2 * 5 * 131
1311 = 3 * 19 * 23
1334-1335 2
1334 = 2 * 23 * 29
1335 = 3 * 5 * 89
1406-1407 2
1406 = 2 * 19 * 37
1407 = 3 * 7 * 67
1434-1435 2
1434 = 2 * 3 * 239
1435 = 5 * 7 * 41
1442-1443 2
1442 = 2 * 7 * 103
1443 = 3 * 13 * 37
1462-1463 2
1462 = 2 * 17 * 43
1463 = 7 * 11 * 19
1490-1491 2
1490 = 2 * 5 * 149
1491 = 3 * 7 * 71
1505-1506 2
1505 = 5 * 7 * 43
1506 = 2 * 3 * 251
1533-1534 2
1533 = 3 * 7 * 73
1534 = 2 * 13 * 59
1581-1582 2
1581 = 3 * 17 * 31
1582 = 2 * 7 * 113
1598-1599 2
1598 = 2 * 17 * 47
1599 = 3 * 13 * 41
1605-1606 2
1605 = 3 * 5 * 107
1606 = 2 * 11 * 73
1614-1615 2
1614 = 2 * 3 * 269
1615 = 5 * 17 * 19
1634-1635 2
1634 = 2 * 19 * 43
1635 = 3 * 5 * 109
1729-1730 2
1729 = 7 * 13 * 19
1730 = 2 * 5 * 173
1742-1743 2
1742 = 2 * 13 * 67
1743 = 3 * 7 * 83
1833-1834 2
1833 = 3 * 13 * 47
1834 = 2 * 7 * 131
1885-1887 3
1885 = 5 * 13 * 29
1886 = 2 * 23 * 41
1887 = 3 * 17 * 37
1946-1947 2
1946 = 2 * 7 * 139
1947 = 3 * 11 * 59
2013-2015 3
2013 = 3 * 11 * 61
2014 = 2 * 19 * 53
2015 = 5 * 13 * 31
2054-2055 2
2054 = 2 * 13 * 79
2055 = 3 * 5 * 137
2085-2086 2
2085 = 3 * 5 * 139
2086 = 2 * 7 * 149
2093-2094 2
2093 = 7 * 13 * 23
2094 = 2 * 3 * 349
2109-2110 2
2109 = 3 * 19 * 37
2110 = 2 * 5 * 211
2134-2135 2
2134 = 2 * 11 * 97
2135 = 5 * 7 * 61
2162-2163 2
2162 = 2 * 23 * 47
2163 = 3 * 7 * 103
2265-2266 2
2265 = 3 * 5 * 151
2266 = 2 * 11 * 103
2289-2290 2
2289 = 3 * 7 * 109
2290 = 2 * 5 * 229
2337-2338 2
2337 = 3 * 19 * 41
2338 = 2 * 7 * 167
2354-2355 2
2354 = 2 * 11 * 107
2355 = 3 * 5 * 157
2378-2379 2
2378 = 2 * 29 * 41
2379 = 3 * 13 * 61
2397-2398 2
2397 = 3 * 17 * 47
2398 = 2 * 11 * 109
2405-2406 2
2405 = 5 * 13 * 37
2406 = 2 * 3 * 401
2409-2410 2
2409 = 3 * 11 * 73
2410 = 2 * 5 * 241
2485-2486 2
2485 = 5 * 7 * 71
2486 = 2 * 11 * 113
2505-2506 2
2505 = 3 * 5 * 167
2506 = 2 * 7 * 179
2585-2586 2
2585 = 5 * 11 * 47
2586 = 2 * 3 * 431
2634-2635 2
2634 = 2 * 3 * 439
2635 = 5 * 17 * 31
2665-2667 3
2665 = 5 * 13 * 41
2666 = 2 * 31 * 43
2667 = 3 * 7 * 127
2678-2679 2
2678 = 2 * 13 * 103
2679 = 3 * 19 * 47
2685-2686 2
2685 = 3 * 5 * 179
2686 = 2 * 17 * 79
2697-2698 2
2697 = 3 * 29 * 31
2698 = 2 * 19 * 71
2702-2703 2
2702 = 2 * 7 * 193
2703 = 3 * 17 * 53
2714-2715 2
2714 = 2 * 23 * 59
2715 = 3 * 5 * 181
2765-2766 2
2765 = 5 * 7 * 79
2766 = 2 * 3 * 461
2769-2770 2
2769 = 3 * 13 * 71
2770 = 2 * 5 * 277
2794-2795 2
2794 = 2 * 11 * 127
2795 = 5 * 13 * 43
2821-2822 2
2821 = 7 * 13 * 31
2822 = 2 * 17 * 83
2829-2830 2
2829 = 3 * 23 * 41
2830 = 2 * 5 * 283
2914-2915 2
2914 = 2 * 31 * 47
2915 = 5 * 11 * 53
2937-2938 2
2937 = 3 * 11 * 89
2938 = 2 * 13 * 113
2945-2946 2
2945 = 5 * 19 * 31
2946 = 2 * 3 * 491
2954-2955 2
2954 = 2 * 7 * 211
2955 = 3 * 5 * 197
d. One of the four numbers must be a multiple of 4, meaning that 2 appears as a factor at least squared, meaning twice, but the three prime factors by definition should be unique (not repeated).
DefDbl A-Z
Dim crlf$, fct(20, 1)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For n = 2 To 3000
f = factor(n)
If f = 3 Then good = 1 Else good = 0
For i = 1 To f
If fct(i, 1) > 1 Then good = 0: Exit For
Next
If good Then
ct = ct + 1
Else
If ct > 1 Then
Text1.Text = Text1.Text & n - ct & "-" & n - 1 & " " & ct & crlf
For i = n - ct To n - 1
f = factor(i)
Text1.Text = Text1.Text & i & " = "
For j = 1 To f
Text1.Text = Text1.Text & fct(j, 0)
If fct(j, 1) > 1 Then Text1.Text = Text1.Text & "^" & fct(j, 1)
If j < f Then Text1.Text = Text1.Text & " * "
Next
Text1.Text = Text1.Text & crlf
Next
Text1.Text = Text1.Text & crlf
End If
ct = 0
End If
DoEvents
Next n
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function factor(num)
diffCt = 0: good = 1
n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
dv = 2: GoSub DivideIt
dv = 3: GoSub DivideIt
dv = 5: GoSub DivideIt
dv = 7
Do Until dv > limit
GoSub DivideIt: dv = dv + 4 '11
GoSub DivideIt: dv = dv + 2 '13
GoSub DivideIt: dv = dv + 4 '17
GoSub DivideIt: dv = dv + 2 '19
GoSub DivideIt: dv = dv + 4 '23
GoSub DivideIt: dv = dv + 6 '29
GoSub DivideIt: dv = dv + 2 '31
GoSub DivideIt: dv = dv + 6 '37
If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
Loop
If n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
factor = diffCt
Exit Function
DivideIt:
cnt = 0
Do
q = Int(n / dv)
If q * dv = n And n > 0 Then
n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
Else
Exit Do
End If
Loop
If cnt > 0 Then
diffCt = diffCt + 1
fct(diffCt, 0) = dv
fct(diffCt, 1) = cnt
End If
Return
End Function
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Posted by Charlie
on 2018-10-14 17:35:31 |
solution
