[ February 2, 2011 ] [ Last Update January 21, 2013 ]
All numbers above 23 are either Powers or Sums Of Powers
B.S. Rangaswamy (email)

[ November 2011 ]
Combining ninedigital (and pandigital) squares (and cubes) with their
reversals and expressing them as sum of powers
B.S. Rangaswamy - goto second part

Our Beloved Prof V.K Doraswamy - A Memoir (†) - goto memoir

On the topic Sum Of Powers (WONplate 178), I had stated that
" With the liberty to use squares, cubes...
All numbers above 23 are either powers or sums of powers ".
Enclosed is a sheet to substantiate this statement, which may
be of interest to you and other Math lovers.

It is possible to express EVERY NUMBER above 23,
as Sum Of Powers !
To arrive at the least number of constituent
powers is an intellectual task.
Rudiments of this exercise are depicted below.

Evolved readers can think of devising suitable Java script
to instantly arrive at the various constituent powers
for high value numbers.

From B.S. Rangaswamy [ November 19, 2011 ]

A few days back, I came across a moving vehicle bearing reg. no '9441'.
Last number 9441 cajoled me to set out in quest of the constituent
powers of this 4 digiter :

Thus it was possible to arrive also at 2 tier results. This is to illustrate
that all numbers have several sets of such constituent powers. To arrive
at the least number of constituent powers is really an intellectual task.

[ November 2011 ]
Combining ninedigital (and pandigital) squares (and cubes) with their
reversals and expressing them as sum of powers
by B.S. Rangaswamy

With due regards to Mr Albert H Beiler, I have attempted to find
typical sets of constituent powers related to each
of ninedigital & pandigital squares, depicted in your webpage
The Nine Digits Page 2 under the following headings.

A. Reverse of ninedigital squares (30 nos)
B. Reverse of pandigital squares (87 nos)
C. Palindromic twin ninedigitals (30 nos)
D. Palindromic twin pandigitals (87 nos)

No pandigital cubes exist in 10 digit numbers. Only 6 cubes with
all numerals from 0 to 9 exist in eleven digit series !
(Episode 24 'Cubes with all numerals' of my book "Wonders of Numerals" )

E. Reverse of elevendigital cubes (6 nos)
F. Palindromic twin elevendigitals (12 nos)
G. Palindromic 32 digitals (6 nos)
H. Palindromic 63 digitals (10 nos)

Enclosing detailed sums of powers for all the above combinations,
for your kind perusal and considerations for inclusion
in your renowned website World!Of Numbers.

Our Beloved Prof V.K Doraswamy - A Memoir (†)

It was a shocking news to learn about the sad demise
of our beloved Prof VKD on 6th November 2012. He retired
as Professor of Mathematics from J.C. College of Engineering
Mysore. VKD tried to popularise Maths in Schools and
colleges for decades. His Kannada book Vinoda Ganita is a
gem and narrates in detail puzzles in Leelavati, a Sanskrit
compendium on Maths and puzzles in Astronomy by
Bhaskaracharya, the great Indian Mathematician. VKD has
authored many text books in Maths for Schools and Colleges.

Prof VKD served The Indian Institute of World Culture
Bangalore as active Member / Vice President for decades. He
was an adept in Sanskrit, Law, Literature and Recreational
Maths.

Lately I got very close with professor, when he presided
over a session on "Palindromes" conducted by me at the
Esteemed Institute in the Palindromic year '2002'. He was
kind enough to write a valuable Foreword to my maiden
book Wonders of Numerals, in the year 2004.

I have met Prof V.K.Doraswamy on several occasions after
that and the last one was at a Book Exhibition in Bangalore
Palace Grounds in October 2011, when he came with certain
staff of the Institute to select books for addition to the
Institute library. Then he never forgot to ask me "Any thing
New?". I felt really glad to hand over to him the rudiments of
my new finding "Every number above 23 is a sum of powers"
and sought his blessings. This now appears in enhanced form
as WON 179. I never thought that it will be his last physical
blessings. But now I earnestly carry his heavenly blessings for
the rest of my life.

Following is the gist of a typical VKD's Maths session, which
I had the rare opportunity to attend. It was full of
information and astonishment:

To fill four blank spaces in the series 1, 7, 3, 4, 5, 1,  ,  ,  ,  ,
10, 2, - - - - - - is a task assigned by VKD, which boggled the
young brains.

What is Ramanujan's Number. Answer to this would come
out instantaneously from most of those present.

1729 = 1³ + 12³ = 9³ + 10³, sum of two different
set of cubes.

Which is the next number having similar features?

It is 4104 = 2³ + 16³ = 9³ + 15³ was my ready answer.

VKD announced, next such numbers are

64232 = 17³ + 39³ = 26³ + 36³
842751 = 23³ + 94³ = 63³ + 84³

Finally he concluded saying the sum of six different
5th powers is a perfect 5th power!

5⁵ + 10⁵ + 11⁵ + 16⁵ + 19⁵ + 29⁵ = 30⁵