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Old 2013-08-25, 07:56   #1
wsc811
 
Aug 2013

48 Posts
Wink why continued fractions gives one factor for N=(4m+3)(4n+3)

for example N=989=23*43 ,Sqrt[N]={a0;a1,a2,...,2a0}
{n,Q,P,a}
{0,1,0,31}
{1,28,31,2}
{2,13,25,4}
{3,20,27,2}
{4,41,13,1}
{5,5,28,11}
{6,52,27,1}
{7,7,25,8}
{8,4,31,15}
{9,37,29,1}
{10,25,8,1}
{11,28,17,1}
{12,31,11,1}
{13,19,20,2}
{14,35,18,1}
{15,20,17,2}
{16,23,23,2} Q=23 is one factor
{17,20,23,2}
{18,35,17,1}
{19,19,18,2}
{20,31,20,1}
{21,28,11,1}
{22,25,17,1}
{23,37,8,1}
{24,4,29,15}
{25,7,31,8}
{26,52,25,1}
{27,5,27,11}
{28,41,28,1}
{29,20,13,2}
{30,13,27,4}
{31,28,25,2}
{32,1,31,62}

N=43*103=4429

{0,1,0,66}
{1,73,66,1}
{2,60,7,1}
{3,27,53,4}
{4,52,55,2}
{5,39,49,2}
{6,92,29,1}
{7,5,63,25}
{8,117,62,1}
{9,12,55,10}
{10,17,65,7}
{11,89,54,1}
{12,36,35,2}
{13,85,37,1}
{14,25,48,4}
{15,69,52,1}
{16,60,17,1}
{17,43,43,2} Q=43 is one factor
{18,60,43,1}
{19,69,17,1}
{20,25,52,4}
{21,85,48,1}
{22,36,37,2}
{23,89,35,1}
{24,17,54,7}
{25,12,65,10}
{26,117,55,1}
{27,5,62,25}
{28,92,63,1}
{29,39,29,2}
{30,52,49,2}
{31,27,55,4}
{32,60,53,1}
{33,73,7,1}
{34,1,66,132}

in many cases, it's so .and this can be used for decomposition?
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