Universal Journal of Applied Mathematics 3(5): 102-111, 2015
DOI: 10.13189/ujam.2015.030503

http://www.hrpub.org

Generalized Parametric Solutions to Multiple
Sums of Powers
Oliver Couto1,*, Seiji Tomita2
1
University of Waterloo, Canada
Tokyo software company (Inc.), Tokyo, Japan*Corresponding Author: samson@celebrating-mathematics.com, fermat@m15.alpha-net.ne.jp
2

Copyright©2015 by authors, all rights reserved. Authors agree that this article remains permanently open access under theterms of the Creative Commons Attribution License 4.0 International License

Abstract The work on equation (B) below has been studied by others for different degree ‘n’ and equation (A) below for
degree five, has been previously published by Mr. Ajai Choudhry (Ref. no. 1). But combined systematic analysis for degrees
2,3,4,5,6,7,8 & 9 etc. has not been done before, as is done in this paper. C( ) and( )Keywords Sums of Powers, Diophantine Equations, Number Theory, Pure Math

1. Introduction
The equation (A) in this paper has three terms on either side in section one and equation (B) has four terms on both sides insection two. In other words this paper deals with degree’s (2,3,4,5,6,7 & 8) for equation (B) & gives a generalized parametricsolutions for any degree ‘n’ for equation (A). As is known that solving Diophantine equations for power five and above isquite difficult. In this paper we show that equation (A) given above is solvable when (s,t,u,w,x,y) are known and we provide
methods to find integer coefficents (a,b,c) for which the above Diophantine equation is satisfied for degree<10. Similarly,analysis is provided for equation (B) also. This paper thus shows that parametric solutions are possible for the Diophantine
equations (B) for degree n < 10 & gives a generalized parametric solution for equation (A) above for any degree ’n’.

2. Discussion (See Below)
1. Consider Equation(A)
Let S=pt+1, T=qt+1, U=t+1
and W=pt-1, X=qt-1, Y= t-1&c = - (ap+bq)
After substituting in equation (A) and solving for the coefficients (a,b,c) we get:For even degree’s (n = 4, 6 & 8)
n=4 we get :
(()

())

(()())(()()))(For n=6 we get:()(()()()())

()

()
()

For n = 8 we get:
(()

(()

(((())()()()))

(()))

())

Note: Similarly for n=10, 12, 14, etc. similar pattern with the coefficient’s is given in the table below.
For odd degree’s (n = 3, 5, 7 & 9)
n=3 we get(()())(()())(()())For n=5 we get,(()()())(()()())

(()

(()

((()

()

())

For n=7 we get,((()())()()())(())()))())

For n=9 we get,(((()((()))

()

()

()
(())

())

()

())

()

())

Note: Similarly for odd ‘n’ = 11,13,15, etc a similar pattern of the coefficient’s is given, in the attached tables.
Table 1. Pascal’s Triangle
nbcdefghjkLmnpqrstu

1112132141561781918941035563611535126

16217084

15202128

31015716561341156
126

1728
84

1836

191

We have the well known Pascal’s Triangle (given above)
Above is for different degree’s ‘n’:
Above is table of Coefficient’s of expansion of (m+1)^n*Coefficient ( a, b, c, d, e, f, --------u) for () and selected degree ‘n’ is given by,
[() ]

( )
(See below for more tables)
Equation,(A)Table 2. For odd degreeDegree nconstantCoeff.
( )Coeff.
( )Coeff. ( )Coeff.
( )Coeff.
( )31n----

512nn---

713n5nn-

-

914n

14n

84n

-

1115n30n

42n

15nn

1316n55n

132n

99n

22n

Coeff.
( )Coeff.
( )Coeff.
( )Remarksn=prime

n

15

17n

91n

5005

429n

3003

455n

1718n

140n

728n

1430n

1144n

364n

40n

n=primen

Where (a, b, c) are polynomials in (t)For above table, coefficients of ( ) is given by, =n(n-1)/2For degree eleven we get coefficient of ( ) =n*(11-1)/2= 5n
and the coefficients of ( ) is given by =n (n-1)(n-2)(n-3)/24
For degree eleven we get coeff. of ( ) = n*(11-1)*(11-2)*(11-3)/24=30nTable 3. For degree ‘n’ even
Coeff.
(t^4)

Degree nconstant

Coeff. (t^2)

Coeff. (t^6)

Coeff. (t^8)

Coeff. (t^10)

422

6

310

38

4

28

28

410

560

12660

512

6110

396

396

110

614

7

182

1001

1716

1001

182

Coeff. (t^12)

7The constant terms in above table has the relation given below Table (3),K = (n)(n-1)(n-2)/12 for degree 12For degree n=12, the coefficient of constant term is = ( 12*11*10)/12 = 110

Table 4. For odd degree ‘n’
Coefficient’s for polynomial (a) inequation (A) above

Representation for degree ‘n’ = (3,5,7,9, -------u)

Remarks

1
n*(n-1)/( 2! )n(n-1)(n-2)(n-3)/( 4! )n(n-1)(n-2)(n-3)(n-4)(n-5)/( 6! )t^p , (p=8)

n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-p+1)/( p! )-- --------------------------------------------------------[ ( )(Where, ( p! ), represents (p) factorial) ]Same formula as Pascal’stableTable 5. For even degree ‘n’
Coeeficent’s for polynomial(a) in equation (A)above

Representation for degree ‘n’ = (4,6,8,10, -------v)

Remarks

2n(n-1)(n-2)/( 4! )3n(n-1)(n-2)(n-3)(n-4)/( 6! )4n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)/( 8! )5n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)(n-8)/( 10! )[ (p+4)(n)(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n – p - 2)/2] * (( p+4)! )(p=8)-- -----------------------------------------------------[( ) (This representation is similar likethe form given in table (a) above.

) ]Where, ( w! ) represents [(p+1)! ] factorial & ‘n’ is relevant degree of the equation.
For n=9 we get the following equations for the coefficients (a,b,c) of the equation (A) and is given below,(((())

(()()

()

()

((()())(())))(()))())

Putting n=9 we get,(()()))

()

()

()

(()()

(((((())()(())))

Similarly we get:((()))(()(General equation for degree ‘n’ odd is given below,
(()()

( ())
(()())(( ())))

()

()

( ())

And similarly for (b & c)
For even, n = 8 we get:((() (() () ((* ()

* ()

* ()

*())

)

()()

()

(Where (w=p+1)
Hence general solution for even degree ‘n’ is given below,)()))())))))

() ( ()*()( ())()*(* ()For example in the above for n=8 & coefficient of ()( ()))*()())

we get for p=2. w=p+1=2+1=3 ,=((()) (* () ()* ()* ()))))*(* ()))*()* (( * (* ()) ( * ((= 56)* ()))*(Similarly we get,((((()))
(())

(()))

())

Numerical solutions by taking values of variable (p, q, t) = (3,2,2) is given in the below table
()

()

()

()

Table 6. Below for odd degree ’n’ find numerical solutions
Degree ‘n’abcst

uw

x

y

3-23

70-71

7

5

3

5

3

1

5-1199

6478

-9359

7

5

3

5

3

1

7-35783369430-631511

7

5

3

5

3

1

9-947039

19170718-35500319

7

5

3

5

3

1

Section (2)
(B)
()

(())

()

( )

()

(())

(())

(()( ) ( )
()

()())

(C)
(D)

()

()

( )

( )

( )

()( )

( )

( )

( )

(𝑫𝑻𝒉∶)( )

( )
()

(())

(( ))

( )

∶()

()

(({)())})((([][())][(()))()

]

[]

[( )

[]

( )]

[]

[]

[]

[]

[]
()

(()

(()

(()())(()))(()( ))((𝑫()((())())()())( )()()

𝒖()())())((()))

(()}
]

[]

()

()

()
{[)

()

()

Numerical solution is:()

()

()

()

()

()

𝑫
()

(()()𝒊𝒗)

()

()

()

(())

()
{}

[[]

[

((()(()())()([()(()))(())]])))([])(((]()))(())()

[]

[[]]

[]

[]
𝑫

(())

𝒔𝒊𝒙

(())

()

()

()

(())

()

()
{}

[[]

[𝑷𝒂 𝒂𝒎()][]]𝒊𝒄 𝑺 𝒍𝒖 𝒊 𝒏()(([)()])])

([𝑫()

(())

𝑺 𝒗 𝒏()

()

()

()

(()()()
(()))

({[]

[][[((()}]]()()()())()))()
{}

[]

[]

[]

[]
𝑫()

()

()

𝑬𝒊 𝒉()

()

()

()

()

{
[[[]][}
]

[]

]

[]
[]

[]

REFERENCES[1] Ajai Choudhry, Diophantine equation, for fifth degree, Rocky mountain journal of mathematics, vol.29,no.2, (1999)[2] Oliver Couto, Published Math paper, Taxicab Equations for power (2, 3, 4 & 5), Journal, International Math Forum, Hikari ltd., Vol.9,Jan 2014.n0.12, pages 561-577.
[3] Oliver Couto & Seiji Tomita, Published math paper, Solution of Polynomial Equation of any Degree ’n’ with Special Emphasis for n=(2, 3, 4, 5 & 6), Journal “SOP transactions on applied mathematics” ISSN(Print): 2373-8472 ISSN: 2373-8480, July 2015[4] Seiji Tomita, Computation number theory web pages:

http://www.maroon.dti.ne.jp/fermat[5] Oliver Couto, Web page on Mathematics, http://www.celebrating-mathematics.com[6] Seiji Tomita, Fourth power polynomial equation, Computation number theory- webpage, http://www.maroon.dti.ne.jp/fermat/diop122e.html[7] Seiji Tomita, sixth power polynomial equation, Computation number theory- Web-page,
http://www.maroon.dti.ne.jp/fermat/diop160e.html[8] Ramanujan lost notebook, Narosa publishing house[9] Euler Leonhard, Opera Omnia, 1984
[10] Tito Piezas-Online collection of algebraic identities

http://sites.google.com/site/tpiezas[11] Ajai Choudhry, Symmetrical Diophantine equations, Journal of mathematics, Rocky mountain journal vol. 34,no.4,winter(2004),
pg. 1261-1298.

[12] Jaroslaw Wroblewski, Tables ofwww.math.uni.wroc.pl/~jwr/eslp

Numerical

solutions[13] L. E. Dickson, history of the theory of numbers, Vol.II,---------------------------------

fordegreethree,

four,six

seven&nine,

website,(Diophantine analysis), AMS Chelsea publication, reprinted year 2000)