## is it known ? - divisibility for a prime p

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• 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:
Message 1 of 2 , Nov 1, 2012
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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

[Non-text portions of this message have been removed]
• 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 2 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 !

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

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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