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Old 2008-12-18, 06:06   #1
devarajkandadai
 
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Default 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
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Old 2008-12-18, 07:37   #2
wblipp
 
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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
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Old 2008-12-18, 16:39   #3
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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
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Old 2008-12-19, 03:33   #4
devarajkandadai
 
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Default Mally's marginal notes

Tks; I just wanted members to remember him on 18th.
Devaraj
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