PART 12. Ninth Powers (Link 29)

PART 13. Tenth Powers (Link 30)

PART 14. Eleventh and Higher Powers (Link 31)

PART 15. Index of Updates (Link)

Index

Part 0. Introduction

Part 1. Assorted Identities

Part 2. Sums of Squares

Part 3. Quadratic Polynomial as a kth Power

Part 4. Simultaneous Polynomials Made Squares

Part 5. Pell Equations

Part 6. Third Powers

Part 7. Fourth Powers

Part 7b. Equal Sums of Like Powers

Part 8. Fifth Powers

Part 9. Sixth Powers

Part 10. Seventh Powers

Part 11. Eighth Powers

Part 12. Ninth Powers

Part 13. Tenth Powers

Part 14. Eleventh and Higher Powers

Part 15. Index of Updates

1) January 7) July

2) February 8) August

3) March 9) September (see Articles 1-3)

4) April 10) October (see Articles 4-6)

5) May (none) 11) November (see Article 7)

6) June 12) December (see Articles 8-11)

(Pls read first: This almost 300-page book is divided into more than 30 sections. For navigation, note that the topic with a link, and those immediately below it without a link, belong to the same section. Alternatively, one can use the Home and other buttons on the sidebar. For questions and comments, feel free to email author at tpiezas@gmail.com.)

Part 0. Introduction (Link 0)

Part 1. Assorted Identities (Link 1)

Part 2. Sums of Squares (Link 2)

I. Sums of two squares

1. x²+y² = z^k

2. x²+ny² = z^k; ax²+by² = cz² (Link 3)

3. ad-bc = ±1

4. x²+y² = z²+1

5. x²+y² = z²-1

6. x²+y² = z²+nt²

7. x²+y² = z²+t^k

8. x²+y² = mz²+nt²

9. c₁(x²+ny²) = c₂(z²+nt²)

10. mx²+ny² = mz²+nt²

II. Sums of three squares (Link 4)

1. x²+y²+z² = t^k

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)Poly(t)

6. x²+y²+z² = 3xyz

III. Sums of four or five squares (Link 5)

1. a²+b²+c²+d² = e^k

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

IV. Some Identities of Squares (Link 5b)

1. Euler-Aida Ammei Identity

2. Brahmagupta-Fibonacci Two-Square Identity

3. Euler Four-Square Identity

4. Degen-Graves-Cayley Eight-Squares Identity

5. V. Arnold’s Perfect Forms

6. Lagrange’s Identity

7. Difference of Two Squares Identity

Part 3. Quadratic Polynomial as a kth Power (Link 6)

I. Univariate: ax²+bx+c = z²

II. Bivariate: ax²+bxy+cy² = z^k

1. x²+cy² = z^k

2. ax²+cy² = z^k, k odd

3. x²+2bxy+cy² = z^k

4. ax²+2bxy+cy² = z^k, k odd

III. Bivariate: ax²+bxy+cy² = dz²

• 1. ax²+bxy+cy² = dz²

  2. ax²+by²+cz²+dxy+exz+fyz = 0

  3. ax²+cy² = dz^k, k > 2

PART 4. Simultaneous Polynomials Made Squares (Link 7)

I. Two variables

1. {x²+axy+by², x²+cxy+dy²}

2. {x²-ny², x²+ny²}

3. {x²+y, x+y²}

4. {x²+y²-1, x²-y²-1}

5. {x²+y²+1, x²-y²+1}

II. Three variables

1. {x ± y, x ± z, y ± z}

2. {x²-y², x²-z², y²-z²}

3. {x²+y², x²+z², y²+z²}

4. {x²+y²+z², x²y²+x²z²+y²z²}

5. {-x²+y²+z², x²-y²+z², x²+y²-z²}

6. {2x²+y²+z², x²+2y²+z², x²+y²+2z²}

7. {2x²+2y²-z², 2x²-y²+2z², -x²+2y²+2z²}

8. {x²+yz, y²+xz, z²+xy}

9. {x²+y²+xy, x²+z²+xy, y²+z²+xy}

10. {x²-xy+y², x²-xz+z², y²-yz+z²}

11. {x²+axy+y², x²+bxz+z², y²+cyz+z²}

III. Four variables

1. {a²+b²+c², a²+b²+d², a²+c²+d², b²+c²+d²}

2. {a²b²+c²d², a²d²+b²c²}

3. {a²b²+c²d², a²c²+b²d², a²d²+b²c²}

4. {1+abc, 1+abd, 1+acd, 1+bcd}

PART 5. Pell Equations (Link 8)

I. Complete Solution

II. Transformations

III. Polynomial Parametrizations

IV. Diophantine Equations needing Pell Equations (Link 9)

PART 6. Third Powers (Link 10)

I. Sums of cubes

1. x³+y³ = z³

2. x³+y³+z³+t³ = 0

3. x³+y³+z³ = 1

4. x³+y³+z³ = 2

5. x³+y³+z³ = (z+m)³

6. p(p²+bq²) = r(r²+bs²)

7. (x+c₁y)(x²+c₂xy+c₃y²)^k = (z+c₁t)(z²+c₂zt+c₃t²)^k

8. x³+y³+z³ = at³ (Link 11)

9. x³+y³ = 2(z³+t³)

10. w³+x³+y³+z³ = nt³

11. x³+y³+z³ = t²

12. x^k+y^k+z^k = {p², q³}, k =2,3

13. x^k+y^k+z^k = t^k+u^k+v^k, k = 1,3

14. x^k+y^k+z^k = t^k+u^k+v^k, k = 2,3

15. x³+y³+z³ = 3t³-t

16. x³+y³+z³ = m(x+y+z)

17. x₁^k+x₂^k+x₃^k+x₄^k = y₁^k+y₂^k, k = 1,2,3

18. x₁^k+x₂^k+x₃^k+x₄^k = y₁^k+y₂^k+y₃^k+y₄^k, k = 1,2,3

19. ax³+by³+cz³ = N (Link 12)

20. ax³+by³+cz³+dxyz = 0

II. Cubic polynomials as kth powers

A. Univariate: ax³+bx²+cx+d² = t^k

1. ax³+bx²+cx+d² = t²

2. ax³+bx²+cx+d² = t³

B. Bivariate: ax³+bx²y+cxy²+dy³ = t^k

1. x³+y³ = t²

2. ax³+by³ = t²

3. x³+y³ = nz²

4. x³+ax²y+bxy²+cy³ = t²

5. x³+ax²y+bxy²+cy³ = t³

PART 7. Fourth Powers (Link 13)

I. Sums of biquadrates

1. a⁴+b⁴ = c⁴+d⁴

2. pq(p²+q²) = rs(r²+s²)

3. pq(p²-q²) = rs(r²-s²)

4. pq(p²+hq²) = rs(r²+hs²)

5. x⁴+y⁴ = z⁴+nt²

6. x⁴+y⁴ = z⁴+nt⁴

7. u⁴+nv⁴ = (p⁴+nq⁴)w²

8. u⁴+nv⁴ = x⁴+y⁴+nz⁴

9. u⁴+v⁴ = x⁴+y⁴+nz⁴

10. x⁴+y⁴+z⁴ = t⁴ (Link 14)

11. x⁴+y⁴+z⁴ = nt^k

12. a^k+b^k+c^k = d^k+e^k+f^k, k = 2,4

12.1 a+b = nc; d+e = nf (Link 15)

12.2 a+b ≠ c; d+e ≠ f

12.3 a+b±c = n(d+e±f) (Link 16)

12.4 na+b+c = d+e+nf

12.5 na+b = e+nf

12.6 a+d = n(c+f)

12.7 (a²-f²)c² = -(b²-e²)d²

1. a^k+b^k+c^k = 2d^k+e^k, k = 2,4

2. a^k+b^k+c^k = d^k+e^k+f^k, k = 2,3,4

3. x⁴+y⁴+z⁴ = 2(x²y²+x²z²+y²z²)-t² (Link 17)

4. v⁴+x⁴+y⁴+z⁴ = nt^k

5. v^k+x^k+y^k+z^k = a^k+b^k+c^k+d^k, k = 2,4

6. 2(v⁴+x⁴+y⁴+z⁴) = (v²+x²+y²+z²)²

7. x₁^k+x₂^k+x₃^k+x₄^k+x₅^k = y₁^k+y₂^k+y₃^k, k = 1,2,3,4

8. x₁^k+x₂^k+x₃^k+x₄^k+x₅^k = y₁^k+y₂^k+y₃^k+y₄^k+y₅^k, k = 1,2,3,4

9. x₁⁴+x₂⁴+…x_n⁴, n > 4

II. Quartic Polynomials as kth Powers (Link 18)

1. ax⁴+by⁴ = cz²

2. ax⁴+bx²y²+cy⁴ = dz²

3. au⁴+bu²v²+cv⁴ = ax⁴+bx²y²+cy⁴

4. ax⁴+bx³y+cx²y²+dxy³+ey⁴ = z²

PART 8. Fifth Powers

I. General Conjectures and Problems (Link 19)

II. Some Theorems on Equal Sums of Like Powers

III. Fifth Powers (Link 20)

5.1 Four terms

5.2 Six terms

5.3 Seven terms (Link 21)

5.4 Eight terms (Link 22)

5.5 Ten terms

5.6 Twelve terms

PART 9. Sixth Powers (Link 23)

6.1 Four terms

6.2 Six terms

6.3 Seven terms (Link 24)

6.4 Eight terms (Link 25)

PART 10. Seventh Powers (Link 26)

7.1 Eight terms

7.2 Nine terms

7.3 Ten terms (Link 27)

PART 11. Eighth Powers (Link 28)

8.1 Eight terms

8.2 Nine terms

8.3 Ten terms

8.4 Twelve terms

8.5 Fourteen terms

8.6 Sixteen terms

8.7 Seventeen terms