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A Collection of Algebraic Identities

By T. Piezas III
 
 
“Everything has beauty, but not everyone can see it.”  - Confucius
 
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 8. Fifth Powers

Part 9. Sixth Powers

Part 10. Seventh Powers
Part 11. Eighth Powers
Part 12. Ninth and Higher Powers
 
 
(Pls read first:  This 200+ page book is divided into 29 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 = zk
  2. x2+ny2 = zk;  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+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

 

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

 

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

 

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. 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
  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 = 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)

 

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 = 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

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    Ten terms

8.3    Fourteen terms

8.4    Sixteen terms

 

PART 12. Ninth Powers and Higher

  
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P.S.  If you like the theory of equations or some of Ramanujan's basic work, see also my sites: A Page On Solvable Equations and The Ramanujan Pages.
 
 
© 2009.  T. Piezas III  (tpiezas@gmail.com)