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

It works because 100[sup]2[/sup] is -1 (mod 137).

By construction 100a + b = 0 (mod 137)

Hence

100a = -b (mod 137)

squaring

10000a[sup]2[/sup] = b[sup]2[/sup] (mod 137)

-a[sup]2[/sup] = b[sup]2[/sup] (mod 137)

0 = a[sup]2[/sup] + b[sup]2[/sup] (mod 137)


The same kind of trick will work for any factor of 10[sup]2n[/sup]+1. So exactly the same trick will work for 73. ie

73*14=1022; 10[sup]2[/sup]+22[sup]2[/sup]= 584 = 8*73

9901 divides 1000[sup]2[/sup]+1. so

9901*8 = 79208; 79[sup]2[/sup]+208[sup]2[/sup]=49505 = 5*9901

The trick extends to other powers, too. For x[sup]3[/sup] you need factors of 10[sup]3n[/sup]-1. 37 is the largest factor for n=2, so we get

29*37 = 1073; 10[sup]3[/sup]+73[sup]3[/sup] = 390017 = 10441*37

11*37=407; 4[sup]3[/sup]+7[sup]3[/sup] = 407 = 11*37

42*37=1554; 15[sup]3[/sup]+54[sup]3[/sup]=160839 = 4347*37

37 is also a factor for n=3 (and higher). So the same 37 also gives

29*37 = 1073; 1[sup]3[/sup]+73[sup]3[/sup] = 389018 = 10514*37

42*37=1554; 1[sup]3[/sup]+554[sup]3[/sup]=170031465 = 4595445*37

Mally's marginal notes
 

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