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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)

Articles by Oliver Couto:

10) Sum of four ( n th ) powers"

11) Equal sums of four nth powers

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

i) Quintic Equation

j) Paper on xyz(x+y+z)

k) Paper, (a^4+b^4)=n(c^4+d^4)

l) Paper, Degree Sixteen paper

m) Paper, Equal sums of 3,4,5,6 quartics

n) Paper, Two 6th powers equals difference of two 4th powers

o) Sums of three fourth powers a multiple of nth 4th powers

p) Sum of Biquadratics with integer coeeficents

q) Solutions, (pa^n+qb^n=rc^n), n=2,3,4,5,6

r) Paper on equal sums of sixth powers with (w+1) terms

s) Diophantine quintic equation with equal sums of 2p & 2q terms

t) Sum of three nth powers equal to a Square

u) Sums of three & four (nth) powers

V) Sixth degree diophantine polynomial equation