Mally's marginal notes

[devarajkandadai]

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

[wblipp]

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

[wblipp]

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

[devarajkandadai]

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