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 2008-12-18, 06:06 #1 devarajkandadai     May 2004 1001111002 Posts Mally's marginal notes In a book on Einstein - co-authored by many well-known physicists, Mally had noted that 137 ( reciprocal of the constant of fine structure) has a peculiar property: when multiplied by a natural number, say 8, we get 1096; 10^2 + 96^2 = 9316=68*137. Many products of natural numbers and 137 have similar properties. a) Is there a simple algebraic explanation for this? b) Are there other numbers with similar properties? A.K. Devaraj
 2008-12-18, 07:37 #2 wblipp     "William" May 2003 New Haven 3·787 Posts It works because 1002 is -1 (mod 137). By construction 100a + b = 0 (mod 137) Hence 100a = -b (mod 137) squaring 10000a2 = b2 (mod 137) -a2 = b2 (mod 137) 0 = a2 + b2 (mod 137) The same kind of trick will work for any factor of 102n+1. So exactly the same trick will work for 73. ie 73*14=1022; 102+222= 584 = 8*73 9901 divides 10002+1. so 9901*8 = 79208; 792+2082=49505 = 5*9901
 2008-12-18, 16:39 #3 wblipp     "William" May 2003 New Haven 3·787 Posts The trick extends to other powers, too. For x3 you need factors of 103n-1. 37 is the largest factor for n=2, so we get 29*37 = 1073; 103+733 = 390017 = 10441*37 11*37=407; 43+73 = 407 = 11*37 42*37=1554; 153+543=160839 = 4347*37 37 is also a factor for n=3 (and higher). So the same 37 also gives 29*37 = 1073; 13+733 = 389018 = 10514*37 42*37=1554; 13+5543=170031465 = 4595445*37
 2008-12-19, 03:33 #4 devarajkandadai     May 2004 22×79 Posts Mally's marginal notes Tks; I just wanted members to remember him on 18th. Devaraj

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