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Web pages: (Maintained by) Oliver Couto

Email:samson@celebrating-mathematics.com

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PARAMETRIC EQUATIONS FOR DEGREE TWO, THREE, FOUR, FIVE, SIX & SEVEN



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Part 1. Misc. Identities Degree 2,3,4,5,6 & 7 (To be posted)

Part 2. Sums of Squares

Part 3. Third Powers

Part 4. Fourth Powers

Part 5. Fifth Powers

Part 6. Sixth Powers

Part 7. Seventh Powers

 

 

Part 1. Well known Miscellanous Identities: (Degrees 2,3,4,&5)

 

 

Second degree

a = n2 -m2, b=2mn, c=m2+n2

 

Third degree

3m2+5mn-5n2, 4m2-4mn+6n2, 5m2-5mn-3n2; 6m2-4mn+4m2

 

Fourth degree

4-1-5 parametric solution

(2x2+12xy-6y2)4 + (2x2-12xy-6y2)4 + (4x2-12y2)4 + (4x2+12y2)4 + (3x2+9y2)4 = 54(x2+3y2)4

 

 Fifth degree

((75y5-x5), (x5+25y5), (x5-25y5), (10x3y2), (50xy4))^5= (x5+75y5)^5

 

Catogeries of Equations (For future Postings)

 

Sum of two squares

1. x2+y2 = zk

2. x2+y2 = z2+1

3. x2+y2 +1= z2

4. x2+y2 = z2+nt2

5. x2+y2 = z2+tk

6. x2+y2 = mz2+nt2

7. c1(x2+ny2) = c2(z2+nt2)

9. mx2+ny2 = mz2+nt2

 

 

  Sums of three squares 

1. x2+y2+z2 = tk

2. x2+y2+z2 = u2+v2

3. (x2-1)(y2-1) = (z2-1)2

4. x2+y2+z2 = u2+v2+w2

5. x2+y2+z2 = (u2+v2+w2)

6. x2+y2+z2 = 3xyz

 

 

  Sums of four or five squares 

1. a2+b2+c2+d2 = ek

2. a2+b2+c2+d2 = e2+f2

3. a2+b2+c2+d2 = e2+f2+g2

4. a2+b2+c2+d2 = e2+f2+g2+h2

5. a2+b2+c2+d2+e2 = f2

 

 

Sums of cubes 

 

 

1. x3+y3 = z3

 

2. x3+y3+z3+t3 = 0

 

5. x3+y3+z3 = (m)3

 

6. x3+y3+z3 = (w)3

 

7. x3+y3 = 2(z3+t3)

 

8. w3+x3+y3+z3 = nt3

 

9. x3+y3+z3 = t2

 

10. xk+yk+zk = {p2, q3}, k =2,3

 

11. xk+yk+zk = tk+uk+vk,  k = 1,3

 

12. xk+yk+zk = tk+uk+vk,  k = 2,3

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Sum of quartics:

 

1. a4+b4 = c4+d4

 

2. x4+y4 = z4+nt4

 

3. u4+nv4 = x4+y4+nz4

 

4. u4+v4 = x4+y4+nz4

 

5. x4+y4+z4 = ntk

 

6. ak+bk+ck = dk+ek+fk,  k = 2,4

 

7 . ak+bk+ck = 2dk+ek,  k = 2,4

8. ak+bk+ck = dk+ek+fk,  k = 2,3,4

9. v4+x4+y4+z4 = ntk

10. v4+x4+y4+z4 = w4

11. vk+xk+yk+zk = ak+bk+ck+dk,  k = 2,4

12. 2(v4+x4+y4+z4) = (v2+x2+y2+z2)2

13. x1k+x2k+x3k+x4k+x5k = y1k+y2k+y3k, k = 1,2,3,4

14. x1k+x2k+x3k+x4k+x5k = y1k+y2k+y3k+y4k+y5k, k = 1,2,3,4

15. x14+x24+xm4=w^4 (m greater than 4)

 

 

Index of updates:

Articles by Oliver Couto:

1) Extention of parametric (taxicab) equations ,degrees 2 & 4
2) Various Identities of Interest
3) Six Variables Taxi Cab Part1
4) Equation p^n+q^n = r^n+s^n
5) Six Variables Taxi Cab Part2
6) Equation pa^6+qb^6=rc^6
7) The Title will be "Various identities"
8) Sums of powers (degree 2,3,4,5 &6)
9) "Quintic Equation"

10) Sum of four ( n th ) powers"

11) Equal sums of four nth powers

12) Sums of fourth powers & second powers equal to zero



Published Math Papers by Oliver Couto:

a) Taxicab equations
b) Polynomial Equations
c) Generalized Parametric Solutions to Multiple Sums of Powers
d) Sum of Three Quartics
e) Methods for solving (K+3) & (K+5) biquadratcs
f) Parametric Solutions to sums of six nth powers
g) Parametric solutions to equation pa^n+qb^n=rc^n+rd^n

WEB LINKS:

1) Seiji Tomita, Generalized Taxicab number, Computation number theory-
webpage
,
2) Jaroslaw Wroblewski ,tables of Numerical solutions for 3-1-3 & 4-1-4 equations.
Website
,
3) Tito Piezas- Online collection of algebraic identities,