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Re: [PrimeNumbers] is it known ? - divisibility for a prime p

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  • Makoto Kamada
    Method p|B*c-/+d, (c,p)=1, N=B*a+b -- p|N iff p|a*d+/-b*c Proof N*c-/+(a*d+/-b*c) =(B*a+b)*c-/+(a*d+/-b*c) =B*a*c+b*c-/+a*d-b*c =B*a*c-/+a*d =a*(B*c-/+d)
    Message 1 of 2 , Nov 2, 2012
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      Method
      p|B*c-/+d, (c,p)=1, N=B*a+b --> p|N iff p|a*d+/-b*c

      Proof
      N*c-/+(a*d+/-b*c)
      =(B*a+b)*c-/+(a*d+/-b*c)
      =B*a*c+b*c-/+a*d-b*c
      =B*a*c-/+a*d
      =a*(B*c-/+d)
      p|B*c-/+d --> p|N*c-/+(a*d+/-b*c)
      (c,p)=1 --> p|N iff p|a*d+/-b*c

      On 2012/11/02 5:26, Norman Luhn wrote:
      > 127 times 1651 ?
      > 165*7-12*1=1143, 114*7-3*12=762, 76*7-12*2=508, 50*7-8*12=254,
      > 25*7-4*12=127 , yes !

      Your choice
      127|10*12+7 --> 165*7-1*12=1143, 114*7-3*12=762, 76*7-2*12=508, 50*7-8*12=254, 25*7-4*12=127

      Shorter
      127|10*13-3 --> 165*3+1*13=508, 50*3+8*13=254, 25*3+4*13=127

      Shortest
      127|10*38+1 --> 165-1*38=127

      Reference in Japanese
      http://homepage2.nifty.com/m_kamada/di200112.htm#16

      Makoto Kamada


      On 2012/11/02 5:26, Norman Luhn wrote:
      > Formula for divisibility for a prime p. Maybe it can help to findfactors
      > of a number n ?.
      >
      > I have found a way to find easy a formula for each prime p.
      >
      > Examples:
      >
      > p=7 ,
      >
      > 7 times 1162 ?
      >
      > 116-2*2=112 , 11-2*2=7 , yes !
      >
      > p=13,13=3*4+1
      >
      > 13 time 72956 ?
      >
      > 7295+6*4=7319 , 731+9*4=767, 76+ 7*4=104, 10+4*4=26 , yes !
      >
      > p=41 , 4-1*4=0
      >
      > 41 times 11111 ?
      >
      > 1111-1*4=1107, 110-7*4=82, yes !
      >
      > Today I found this. p=127 is not so easy : 127 is not x*7+12 , but we
      > take 12*7-7*12
      >
      > 127 times 1651 ?
      >
      > 165*7-12*1=1143, 114*7-3*12=762, 76*7-12*2=508, 50*7-8*12=254,
      > 25*7-4*12=127 , yes !
      >
      >
      >
      > --
      >
      > Norman
      >
      >
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