| 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 |
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Catogeries of Equations (For future Postings)
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Sum
of two squares
1.
x2+y2 = zk
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2.
x2+y2 = z2+1
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3.
x2+y2 +1= z2
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4.
x2+y2 = z2+nt2
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5.
x2+y2 = z2+tk
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6.
x2+y2 = mz2+nt2
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7.
c1(x2+ny2) = c2(z2+nt2)
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9.
mx2+ny2 = mz2+nt2
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Sums of three squares
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1.
x2+y2+z2 = tk
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2.
x2+y2+z2 = u2+v2
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3.
(x2-1)(y2-1) = (z2-1)2
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4.
x2+y2+z2 = u2+v2+w2
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5.
x2+y2+z2 = (u2+v2+w2)
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6.
x2+y2+z2 = 3xyz
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Sums
of four or five squares
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1.
a2+b2+c2+d2 = ek
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2.
a2+b2+c2+d2 = e2+f2
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3.
a2+b2+c2+d2 = e2+f2+g2
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4.
a2+b2+c2+d2 = e2+f2+g2+h2
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5.
a2+b2+c2+d2+e2 = f2
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Sums
of cubes
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1. x3+y3
= z3
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2. x3+y3+z3+t3
= 0
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5. x3+y3+z3
= (m)3
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6. x3+y3+z3
= (w)3
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7. x3+y3
= 2(z3+t3)
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8. w3+x3+y3+z3
= nt3
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9. x3+y3+z3
= t2
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10. xk+yk+zk = {p2,
q3}, k =2,3
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11. xk+yk+zk = tk+uk+vk, k =
1,3
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12. xk+yk+zk = tk+uk+vk, k =
2,3
---------------------------------------
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Sum
of quartics:
1.
a4+b4 = c4+d4
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2.
x4+y4 = z4+nt4
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3.
u4+nv4 = x4+y4+nz4
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4.
u4+v4 = x4+y4+nz4
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5.
x4+y4+z4 = ntk
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6.
ak+bk+ck = dk+ek+fk, k =
2,4
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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)
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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
h) Sums of quartics in the Richmond equation
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
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,
4) Ajai Choudhry web site:,
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