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A Collection of Algebraic Identities

Everything has beauty, but not everyone can see it.”  - Confucius

When the land is barren, plant trees.”  - Jean Giono

Index

Part 0. Introduction

Part 2. Sums of Squares

Part 3. Quadratic Polynomial as a kth Power

Part 8. Fifth Powers

Part 9. Sixth Powers

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 1. Assorted Identities  (Link 1)

Part 2. Sums of Squares  (Link 2)

I. Sums of two squares
1. x2+y2 = zk
2. x2+ny2 = zk;  ax2+by2 = cz2   (Link 3)
4. x2+y2 = z2+1
5. x2+y2 = z2-1
6. x2+y2 = z2+nt2
7. x2+y2 = z2+tk
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 = 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)Poly(t)
6. x2+y2+z2 = 3xyz
III. Sums of four or five squares  (Link 5)
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
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

I. Univariate: ax2+bx+c = z2

II. Bivariate: ax2+bxy+cy2 = zk

1. x2+cy2 = zk
2. ax2+cy2 = zk, k odd
3. x2+2bxy+cy2 = zk
4. ax2+2bxy+cy2 = zk, k odd

III. Bivariate: ax2+bxy+cy2 = dz2

1. ax2+bxy+cy2 = dz2
2. ax2+by2+cz2+dxy+exz+fyz = 0
3. ax2+cy2 = dzk, k > 2

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. xk+yk+zk = {p2, q3}, k =2,3
13. xk+yk+zk = tk+uk+vk,  k = 1,3
14. xk+yk+zk = tk+uk+vk,  k = 2,3
15. x3+y3+z3 = 3t3-t
16. x3+y3+z3 = m(x+y+z)
17. x1k+x2k+x3k+x4k = y1k+y2k,  k = 1,2,3
18. x1k+x2k+x3k+x4k = y1k+y2k+y3k+y4k,  k = 1,2,3
20. ax3+by3+cz3+dxyz = 0

II. Cubic polynomials as kth powers

A. Univariate: ax3+bx2+cx+d2 = tk

1. ax3+bx2+cx+d2 = t2
2. ax3+bx2+cx+d2 = t3

B. Bivariate: ax3+bx2y+cxy2+dy3 = tk

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)

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 = ntk
12. ak+bk+ck = dk+ek+fk,  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. ak+bk+ck = 2dk+ek,  k = 2,4
2. ak+bk+ck = dk+ek+fk,  k = 2,3,4
3. x4+y4+z4 = 2(x2y2+x2z2+y2z2)-t2   (Link 17)
4. v4+x4+y4+z4 = ntk
5. vk+xk+yk+zk = ak+bk+ck+dk,  k = 2,4
6. 2(v4+x4+y4+z4) = (v2+x2+y2+z2)2
7. x1k+x2k+x3k+x4k+x5k = y1k+y2k+y3k, k = 1,2,3,4
8. x1k+x2k+x3k+x4k+x5k = y1k+y2k+y3k+y4k+y5k, k = 1,2,3,4
9. x14+x24+…xn4, 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

5.1    Four terms

5.2    Six terms

5.5    Ten terms

5.6    Twelve terms

PART 9. Sixth Powers  (Link 23)

6.1    Four terms

6.2    Six terms

PART 10. Seventh Powers  (Link 26)

7.1    Eight terms

7.2    Nine terms

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

PART 12. Ninth Powers  (Link 29)

PART 13. Tenth Powers  (Link 30)

PART 14. Eleventh and Higher Powers  (Link 31)