“In this article a general expression has been conjectured which gives the number `x` being the maximum number of n th powers required to represent, when added together,any finite integer. In other words, there is no positive finite integer which cannot be represented as a sum of `x` nth powers.“.

Study and research done during summer of 1959 while studying for Post graduate course in Mathemetics in Bombay University.

Article first written in 1960 …not published

Article updated in 1986, and sent with a covering letter to `The Director…Institute of Mathematical Sciences… Madras on 4th April 1986… It was not published but an encouraging response received.

Article recomposed in November 2014 and tells of my letter to IMS Madras, the manuscript of the 1986 article, and compares it with the latest inputs available on `Google` on the subject.The November 2014 article is being published in the website..`www.sixwords.in`

In April 1986, I wrote a letter to The Director, Indian Institute of Mathemetical Sciences, Madras on the subject `Edward Warings conjectures (1770) on theory of numbers, inviting his kind attention to an article in Illustrated Weekly of India, relating to the recent proof of Waring limit for 4th powers by Dr. R. Balasubramanium in collaboration with two French Mathemeticians. It appeared – with that proof – that Waring limits upto 9th power stood proved, at the time. It was also mentioned – in the weekly – that there is no documented evidence available which shows how Edward Waring arrived at his conjectures 200 years back.

In my letter I mentioned that I had derived a general expression for a number `x` in terms of `n`, such that there is no positive integer existing which cannot be representated as a sum of – not more than – `x` nth powers of integers. The derivation of the general expression was entirely based on known Waring Limits upto the 7th power of which I was aware of at the time. However – I wrote – that the authenticity of the general expression can be a gauged from the table – which I enclosed – which gives the value of `x` as worked out from the general expression for upto `n` = 15, by tallying the same with corresponding Waring limits upto 15th power which should be available with the latest literature on the subject. The importance of the general expression can be judged from the fact that if this is proved, it will act as a comprehensive proof of Warings Conjectures for all powers.

The referred table which constituted a part of the letter gave the values `x` for all values of `n` from 2 to 15, as 4, 9, 19, 37, 73, 143, 279, 558, 1079, 2132, 4223, 8384, 16673, and 33203 respectively.

MANUSCRIPT OF THE ARTICLE OF APRIL 1986 :

“Waring limits postulated over 200 years back (in 1770) by Edward Waring have once again been in the limelight following the recent proof of the Waring limit corresponding to the 4th powers by Dr. R. Balasubramanium of the Institute of Mathemetical Sciences in collaboration with French Mathemeticians Jean Yuves Peshouillers and Francois Press.

What are these Waring limits?

Waring`s theorems or rather conjectures, since he could not prove them, are on the following lines :

- a) Every integer is either a square or it can be represented as a sum of two, three, or upto four squares. In other words no positive finite integer exists which cannot be represented as a sum of four or less than four squares. The number four can be considered as a Waring limit for squares.
- b) Every integer is either a cube or it can be represented as a sum of two, three, four, or upto nine cubes. The number nine can be considered as a Waring limit for cubes.
- c) Every integer is either a fourth power, or it can be represented as a sum of two, three , four..or upto 19 fourth powers. The number 19 can be considered as a Waring limit for 4th powers.
- d) Likewise Waring conjectured that every integer is either a 5th power, a 6th power, a 7th power etc, or it can be represented as a sum of upto 37 5th powers, 73 6th powers, 143 7th powers etc respectively, and the numbers 37, 73, and 143 being regarded as Waring limits for 5th, 6th, and 7th powers respectively.

As can be seen, there is no distinct pattern existing between the Waring limits for various powers ( 4, 9, 19, 37, 73, 143, etc), and todate there is no documentary evidence available which shows how Waring arrived at his postulations. During the last 50 years or so, Mathemeticians have proved Warinhs Theorums upto several powers, the latest one being that of Dr. R. Balasubramaniums proof corresponding to 4th power. However it appears there is no record of a successful research having been carried out for the evaluation of a generalised expression for Waring limits, let alone its proof. The author here has conjectured to derive a general expression based on available Waring limits upto 7th power. Using this expression waring limits upto 15th power are determined and indicated in table `A`.

EVALUATION OF GENERAL EXPRESSION FOR DETERMINING WARING LIMITS :

A detailed analytical study of the Waring limits upto 7th power will reveal the following:

- a) The smallest integer that requires the summation of the maximum number of squares is the number 7, and it requires one square of 2 and 3(*) squares of 1.
- b) The smallest integer that requires the summation of the maximum number of cubes is the number 23, and it requires 2 cubes of 2 and 7(*) cubes of 1.
- c) The smallest integer that requires the summation of the maximum number of 4th powers is the number 79 and it requires four 4th powers of 2 and 15(*) 4th powers of 1.

Proceeding in the same way, it can be found that the smallest integer requiring the summation of the maximum number of 5th powers is the number 223 (requiring six 5th powers of 2 and 31(*) 5th powers of 1), for 6th powers the corresponding number is 703( requiring 10 6th powers of 2 and 63(*) 6th powers one ), and for the 7th powers the corresponding number is 2175 (requiring 16 7th powers of 2 and 127 (*) 7th powers of 1.

The numbers marked with an asterisk contained within brackets (*) ie 3, 7, 15, 31, 63, and 127, are significant, and follow a distinctive pattern, each being one short of 2, one short of cube of 2, one short of 4th power of 2, one short of 5th power of 2, one short of 6th power of 2, and one short of 7th power of 2 respectively.

In each of the above cases it is observed that the smallest integer that requires the maximum number of nth powers is less than the nth power of 3. Denoting this smallest integer as `D`, it is seen that `D` requires the summation of nth powers of 2 and 1 only.

Hence D = a*2^n + b*1^n …. …. …. …. …. (A1)

Where `a` and `b` are constants.

Further it is noticed that `D` is in each case found to be exactly one short of the number containing the maximum number of nth powers of 2 before the nth power of 3.

Hence by definition: D = c*2^n – 1 …. …. …. …. (A2)

Where `c` is a constant giving the maximum number of nth powers of contained in 3^n. ie c is equal to the positive integral value ( quotient only) of the expression ( 3^n/2^n).

For n = 2, c = quotient of the expression (3^2/2^2) = 2, and D = 2*2^2 -1 = 7

For n = 3, c = quotient of the expression (3^3/2^3) = 3, and D = 3*2^3 -1 = 23

For n = 4, c = quotient of the expression (3^4/2^4) = 5, and D = 5*2^4 -1 = 79

For n = 5, c = quotient of the expression (3^5/2^5) = 7, and D = 7*2^5 -1 = 223

For n = 6, c = quotient of the expression (3^6/2^6) = 11, and D = 11*2^6 -1 = 703

For n = 7, c = quotient of the expression (3^7/2^7) = 17, and D = 17*2^7 -1 = 2175

As indicated earlier `D` contains nth powers of 2 and 1 only. Denoting by `x` the maximum number of nth powers required to represent D, it is obvious that x is equal to the sum of the quotient (ie the positive integral value) of the expression (D/2^n) and its remainder `R`.

Ie x = quotient of (D/2^n) + R

From equation A2 D = c*2^n – 1

Hence x = quotient of |( c*2^n – 1)/2^n)| + R

Substituting for c as quotient of |3^n/2^n|

Where | | denotes the positive integral value (quotient only) of the expression within.

We get x = {(2^n*|3^n/2^n| -1)/2^n } + R …. ….. …. (A3)

Which is the general expression for Waring limits.

Where { } denotes the positive integral value (quotient only)of the expression within, and R as the remainder.

Thus it can be stated that every integer is either an nth power or it can be represented as the sum of 2, 3, 4….or at the most `x` nth powers, or in other words there exists no positive integer which cannot be represented as the sum of `x` nth powers, where `x` is given by the expression in (A3).

Based on the expression in (A3) the values of `x` as determined upto 15th powers ie for n= 2, 3, 4 …etc …upto n =15 are 4, 9, 19, 37,73, 143, 279, 548,1079, 2132, 4223, 8384, 16673, and 33203 respectively.

A more simplified expression for `x` than that indicated in (A3) can also be determined.

As per (A1) : D = a*2^n + b*1^n, and x = a + b

As found out earlier, the value of `b` is just 1 short of 2^n, ie b = 2^n – 1 and the smallest integer D contains 1 nth power of 2 short of the maximum number of nth powers of 2 that can be contained in 3^n, ie a = | 3^n/2^n | -1

Hence x = a + b = | 3^n/2^n | + 2^n – 2 …. …. …. (A4)

Where, | | denotes the quotient ( ie the positive integral value ) of the expression within.

Expressions ( A3) and (A4) above, give identical results. This can be checked and verified for all values of n from 2 to 15.

CONCLUSION :

On the basis of known Waring limits upto 7th power, a general expression has been derived for `x` in terms of `n`, such that there is no positive integer that exists, which cannot be represented as a sum of x nth powers. The primary assumption which facilitated the derivation being that the smallest integer `D` that requires the summation of maximum number of nth powers is less than 3^n, and that D is in each case just one short of the number containing the maximum number of nth powers of 2 before 3^n. In the validity of these assumptions lies the proof of the general expression which in turn would prove Waring`s conjectures comprehensively.

…. …. Manuscript ends …. ….

UPDATE ON THE SUBJECT :

Following my brief interaction with Institute of Mathemetical Sciences , Madras (now called Chennai) in connection with the manuscript, I tried hard with my search to find the latest literature on the subject in particular on the general expression, but couldn`t find anything worthwhile, there was no Google at the time. And then, I got busy with work –Design and construction of structures – and later got myself deeply involved in Science and philosophy, and the article was out of mind ever since for the last 28 years. In the interim I along with my team of Structural Engineers received an International award ..`The Bentleys Global Be Inspired Award` for innovations in Structural Engineering for the year 2013, and wrote a book on Science and Philosophy called “SIX WORDS… Seminar held in a parallel universe“, published in January 2014.

Now – in November 2014 – I am including the manuscript in my website, to solicit reader`s comments and thought I should find out the latest on the subject ..Through `Google` of course.

During my search, I found an article on `Recent Progress On Waring`s Theorums And Its Generalisations ` …BY L. E. Dickson.

Its a very lengthy write up… here are two small extracts from the article :

“ On the Ideal limit for Universal Waring Theorum : It has been proved that every positive integer is a sum of 9 cubes, whence 9 is the ideal limit for cubes. For biquadrates (fourth powers) the ideal limit is 19.“

“ The ideal limit for nth powers is I = q + 2^n – 2, where q is the greatest integer less than (3^n/2^n)“

Dickson goes on to with his elaborate study and analysis to discuss various theorems related to the subject pertaining to the Ideal limit for various powers. His analysis is extremely complex and difficult to comprehend. It is also a little bit confusing and has a few errors too; In his table of Ideal limits he gives a value of 558 for the Ideal limit for 9th powers, which according to me is 548 (perhaps it might be a misprint).

I could not find in Dickson`s analysis or anywhere else the simplified methodology to arrive at the general expression put forward by me in my article.

I have prepared an excel sheet for evaluation of `x` ( the ideal limit in Dickson`s nomenclature ) for various values of `n`. Proceeding further the values of `x` for n values 16, 17, 18, 19, and 20 are determined as 66190, 132055, 263619, 526502, and 1051899 respectively.

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