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Find Counterexamples (Posted on 2016-07-06) Difficulty: 3 of 5
There was once a conjecture that every odd number except 1 could be written as the sum of a power of 2 and a prime.

Examples:
3=20+2
99=25+67

Find the counterexamples less than 1000.

Show there are an infinite number of counterexamples.

No Solution Yet Submitted by Jer    
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Some Thoughts computer solution to part 1 Comment 1 of 1
 n     differences: n minus each power of 2 below n (showing lowest prime factor and resulting quotient)

127    126(2*63), 125(5*25), 123(3*41), 119(7*17), 111(3*37), 95(5*19), 63(3*21)
149    148(2*74), 147(3*49), 145(5*29), 141(3*47), 133(7*19), 117(3*39), 85(5*17), 21(3*7)
251    250(2*125), 249(3*83), 247(13*19), 243(3*81), 235(5*47), 219(3*73), 187(11*17), 123(3*41)
331    330(2*165), 329(7*47), 327(3*109), 323(17*19), 315(3*105), 299(13*23), 267(3*89), 203(7*29), 75(3*25)
337    336(2*168), 335(5*67), 333(3*111), 329(7*47), 321(3*107), 305(5*61), 273(3*91), 209(11*19), 81(3*27)
373    372(2*186), 371(7*53), 369(3*123), 365(5*73), 357(3*119), 341(11*31), 309(3*103), 245(5*49), 117(3*39)
509    508(2*254), 507(3*169), 505(5*101), 501(3*167), 493(17*29), 477(3*159), 445(5*89), 381(3*127), 253(11*23)
599    598(2*299), 597(3*199), 595(5*119), 591(3*197), 583(11*53), 567(3*189), 535(5*107), 471(3*157), 343(7*49), 87(3*29)
701    700(2*350), 699(3*233), 697(17*41), 693(3*231), 685(5*137), 669(3*223), 637(7*91), 573(3*191), 445(5*89), 189(3*63)
757    756(2*378), 755(5*151), 753(3*251), 749(7*107), 741(3*247), 725(5*145), 693(3*231), 629(17*37), 501(3*167), 245(5*49)
809    808(2*404), 807(3*269), 805(5*161), 801(3*267), 793(13*61), 777(3*259), 745(5*149), 681(3*227), 553(7*79), 297(3*99)
877    876(2*438), 875(5*175), 873(3*291), 869(11*79), 861(3*287), 845(5*169), 813(3*271), 749(7*107), 621(3*207), 365(5*73)
905    904(2*452), 903(3*301), 901(17*53), 897(3*299), 889(7*127), 873(3*291), 841(29*29), 777(3*259), 649(11*59), 393(3*131)
907    906(2*453), 905(5*181), 903(3*301), 899(29*31), 891(3*297), 875(5*175), 843(3*281), 779(19*41), 651(3*217), 395(5*79)
959    958(2*479), 957(3*319), 955(5*191), 951(3*317), 943(23*41), 927(3*309), 895(5*179), 831(3*277), 703(19*37), 447(3*149)
977    976(2*488), 975(3*325), 973(7*139), 969(3*323), 961(31*31), 945(3*315), 913(11*83), 849(3*283), 721(7*103), 465(3*155)
997    996(2*498), 995(5*199), 993(3*331), 989(23*43), 981(3*327), 965(5*193), 933(3*311), 869(11*79), 741(3*247), 485(5*97)

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For n = 3 To 1000 Step 2
  pwr2 = 1
  found = 0
  Do
    p = n - pwr2
    If prmdiv(p) = p And p <> 1 Then
      found = 1
      Exit Do
    End If
    pwr2 = 2 * pwr2
  Loop Until pwr2 >= n
  If found = 0 Then
    Text1.Text = Text1.Text & n & "   "
    pwr2 = 1
    Do
      p = n - pwr2
      Text1.Text = Text1.Text & Str(p) & "("
      Text1.Text = Text1.Text & prmdiv(p)
      p = p / prmdiv(p)
      Text1.Text = Text1.Text & "*" & p
      Text1.Text = Text1.Text & ")"
      pwr2 = 2 * pwr2
      If pwr2 < n Then Text1.Text = Text1.Text & ","
    Loop Until pwr2 >= n
    Text1.Text = Text1.Text & crlf
  End If
 Next
 
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

Function prmdiv(num)
 Dim n, dv, q
 If num = 1 Then prmdiv = 1: Exit Function
 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
 Loop
 If n > 1 Then prmdiv = n
 Exit Function

DivideIt:
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    prmdiv = dv: Exit Function
   Else
    Exit Do
  End If
 Loop

 Return
End Function

  Posted by Charlie on 2016-07-06 10:12:59
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