ODDWHEEL: Magic Squares, Squares of Squares and Interleaved Sequences

Citric and Isocitric/Alloisocitric acids

Home to the Wheel, Loubère, New Chinese Remainder Theorem and Diophantine Methods


Table of Contents A0
  
Citric Acid Section
  
1. Non-Enzymatic Chemical Conversion of Cis Aconitate to Citrate (Part I)
2. Non-Enzymatic Chemical Synthesis of erythro and threo IsoCitrates(Part II)
3. Dehydration of Citric Acid
4. The Reaction of Maleic Anhydride with Dimethyl Malonate
5. The Catalytic Reaction of 1,1,2,3 Propanetetracarboxylic acid with HOI and H2O2
  
Sierpinski/Cantor Set Fractal, Diophantine Equations, Syllogisms and Monkey Problem
  
1. A Novel Square/Triangle Fractal Generated thru Recursion (Part I)
2. A Novel TriHexagon/Rhombus Fractal Generated thru Recursion (Part II)
3. Novel Octagon Recursive Structures (Part III)
4. Arrangement of Middle Terms in Syllogisms
5. Monkey Coconut Problem - Diophantine Equation by extended Euclidian Algorithm (Part I)
6. New Math Sequence from Monkey/Coconut Diophantine Equation (Part II)
7. New Second Math Sequence to the Monkey/Coconut Diophantine Equation (Part III)
8. A General Solution to the Monkey/Coconut Diophantine Sequences (Part IV)
Table of Contents A0a
Sophie Germain Identities
1. A New General Equation and Sequences for Sophie Germain Type Identities (Part I)
2. A New General Equation for Sophie Germain Type Identities-Primality Testing (Part II)
3. A New General Equation and Sequences for Sophie Germain Type Identities (Part III)
3. A New General Equation and Sequences for Sophie Germain Type Identities (Part IV)
  
Table of Contents A0b
Irrational Sequences and Proofs by Induction (Staircase Sequences)
1. Irrational Numbers from Adjacent Natural Numbers (Part I)
2. Irrational Numbers from Adjacent Natural Numbers (Part IIa)
3. Irrational Numbers from Adjacent Natural Numbers (Part IIb)
4. Irrational Numbers from Adjacent Natural Numbers (Part III)-Staircase
5. Irrational Numbers from Adjacent Natural Numbers (Part IVa)
6. Irrational Numbers from Adjacent Natural Numbers (Part IVb)
7. Irrational Numbers from Adjacent Natural Numbers (Part Va)
8. Irrational Numbers from Adjacent Natural Numbers (Part Vb)
9. Irrational Numbers from Adjacent Natural Numbers (Part VIa)
10. Irrational Numbers from Adjacent Natural Numbers (Part VIb)
11. Irrational Numbers from Adjacent Natural Numbers (Part VIIa)
12. Irrational Numbers from Adjacent Natural Numbers (Part VIIc)
13. Irrational Numbers from Adjacent Natural Numbers (Part VIIb)
14. Irrational Numbers from Adjacent Natural Numbers (Part VIId)
15. Irrational Numbers from Adjacent Natural Numbers (Part VIIIa)-Staircase
16. Irrational Numbers from Adjacent Natural Numbers (Part VIIIb)-Staircase
17. Irrational Numbers from Adjacent Natural Numbers (Part IXa)
17a. Irrational Numbers from Adjacent Natural Numbers (Part VIIa2)
17b. Irrational Numbers from Adjacent Natural Numbers (Part VIIb2)
18. Irrational Numbers from Adjacent Natural Numbers (Part IXb)-Staircase
19. Irrational Numbers from Adjacent Natural Numbers (Part Xa)
19a. Irrational Numbers from Adjacent Natural Numbers (Part VIIc2)
19b. Irrational Numbers from Adjacent Natural Numbers (Part VIId2)
20. Irrational Numbers from Adjacent Natural Numbers (Part Xb)-Staircase
21. Irrational Numbers from Adjacent Natural Numbers (Part XIa)
22. Irrational Numbers from Adjacent Natural Numbers (Part XIb)-Staircase
  
Table of Contents A0c
Alternative to Chinese Remainder Theory
1. New Algorithm Involving Quadratic Residues (Part IVa))
2. New Algorithm Involving Quadratic Residues (Part IVb))
3. New Algorithm Involving Quadratic Residues (Part IVc))
4. New Algorithm Involving Quadratic Residues (Part IVd))
5. Expanded Algorithm Involving Quadratic Residues (Part Ia))
6. Expanded Algorithm Involving Quadratic Residues (Part Ib))
7. Expanded Algorithm Involving Quadratic Residues (Part Ic))
  
Table of Contents A0d
Diophantine Multiples
1. The Diophantine General Equation x2 + D∑(y2) = z2 (Intro)
2. The Diophantine General Equation x2 + Dy2 = z2 (Part I)
2c. Comparison of Known Method with the New Diophantine Method (Part IC)
3. The Diophantine General Equation x2 − Dy2 = z2 (Part II)
4. The Diophantine Equation x2 + 2y2 = z2 (Part III)
5. The Diophantine Equation x2 − 2y2 = z2 (Part IV)
6. The Diophantine Equation x2 + 3y2 = z2 (Part V)
7. The Diophantine Equation x2 − 3y2 = z2 (Part VI)
8. The Diophantine Equation x2 + 5y2 = z2 (Part VII)
9. The Diophantine Equation x2 − 5y2 = z2 (Part VIII)
10. The Diophantine Equation x2 + 4y2 = z2 (Part IX)
  
11. The Diophantine Equation - Quadruples (Part I)
12. The Diophantine Equation - Quadruples (Part II)
13. The Diophantine Equation - Quadruples (Part III)
14. The Diophantine Equation - Quadruples (Part IV)
15. The Diophantine Equation - Quadruples (Part V)
16. The Diophantine Equation - Quadruples (Part VI)
17. The Diophantine Equation - Quintuples/Sextuples (Part X)
18. The Diophantine Equation - Quintuples (Part XI)
  
Pythagorean Multiples
19. Pythagorean Multiples For a Diophantine Equation (Part XII)
20. Method for Calculating Pythagorean Multiples (Part XIII)
21. Sequences of Pythagorean Multiples-Random & Arithmetic Access (Part XIV)
22. Sequences of Pythagorean Multiples-Random & Arithmetic Access (Part XV)
23. Sequences of Pythagorean Multiples-Random & Arithmetic Access (Part XVb)
24. Sequences of Pythagorean Multiples (Part XVc)
25. Sequences of Pythagorean Multiples (Part XVI)
26. Sequences of Pythagorean Multiples (Part XVII)
27. Sequences of Pythagorean Multiples (Part XVIII)
28. Sequences of Pythagorean Multiples Via Random Access (Part XIX)
29. Generation of Pythagorean Triples via Random Access of Known Triples (Part XX)
30. Generating Sequences of Pythagorean Triples(Part XXI)
  
Table of Contents A1a
  
1. The Diophantine Equation 2y2 − x2 = z2 and Magic Squares (Part I)
2. The Diophantine Equation 2y2 − x2 = z2 and Magic Squares (Part II)
3. The Diophantine Equation 2y2 − x2 = z2 and Magic Squares (Part I/IIA) Random Access
  
Table of Contents A1b
  
1. A Novel Method for Multiplying Large Numbers
2. The Column Addition Triangle (CAT) - A Pascal Δ Analogy (Part I)
3. The Column Addition Triangle (CAT) and Polynomials (Part II)
4. Triangles Exhibiting Ascending Diagonal Properties (Part III)
5. Pascal Type Triangles(1,2) (Part I)
6. Pascal Type Triangles(1,3) (Part II)
7. The Fibonacci Type Sequences Triangle and their Ascending Diagonals (Part A)
8. The Ascending Diagonals of a Fibonacci Type Sequences Triangle (Part B)
9. Partition Numbers and Methods for Calculation
10. Congruence of Numbers Raised to Some Power (Part I)
11. Congruence of Numbers Raised to Some Power (Part II)
  
Table of Contents A1c
1. Sequences from Quadratic Residues (Part I)
2. Sequences from Quadratic Residues (Part II)
3. Finite Sequences from Quadratic Residues (Part III)
4. Sequences from Quadratic Residues (Part IV)
  
Table of Contents A1d
The Pellian Equation
1. Finding the Diophantine Equation x2 − Dy2 = ±z2 Given (x,y,z) (Part IX)
2. The Pellian Equation x2 −Dy2 = 1 Revisited (Part I)
3. The Pellian Equation x2 −Dy2 = ±1 Revisited (Part II)
4. The Pellian Equation x2 −Dy2 = ±1 From a Sequence Sn (Part IIIA)
5. The Pellian Equation x2 −Dy2 = ±1 Continuation (Part IIIB)
6. The Pellian Equation x2 −Dy2 = 1 from the Sequence (n+1)2 − 1 (Part IV)
7. The Pellian Equation x2 −Dy2 = 1 from the Sequence n(n+1)(Part V)
8. The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part VI)
9. The Pellian Equation x2 −Dy2 = 1 from a new Paired Sequence P(n) (Part VII)
10. The Pellian Equation x2 −Dy2 = 1 from a Paired Sequences P(n) (Part VIII)
11. The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part IX)
12. The Pellian Equation x2 −Dy2 = 1 from two Paired Sequences (Part XA)
13. The Pellian Equation x2 −Dy2 = 1 from two Paired Sequences (Part XB)
14. The Pellian Equation x2 −Dy2 = 1 from two Paired Sequences (Part XC)
15. The Pellian Equation x2 −Dy2 = 1 from two Paired Sequences (Part XD)
16. The Pellian Equation x2 −Dy2 = 1 from two Paired Sequences (Part XE)
17. The Pellian Equation x2 −Dy2 = 1 from Multiple Sequences (Part XFa)
18. The Pellian Equation x2 −Dy2 = 1 from Multiple Sequences (Part XFb)
19. The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part XI)
20. The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part XIIA)
21. The Pellian Equation x2 −Dy2 = 1 from a Paired Sequence P(n) (Part XIIB)
22. Calculating the Convergents of x2 −Dy2 = ±1 via Programming (Part XIII)
23. The Pellian Equation x2 −Dy2 = ±1 from the Sequence (n + 1)2 + 1 (Part XIV)
24. The Pellian Equation x2 −Dy2 = ±1 from Dual Sequences (Part XV)
25. The Pellian Equation x2 −Dy2 = −1 from a new Sequence Sn (Part XVI)
26. The Pellian Equation x2 −Dy2 = 1 from a Sequence Sn (Part XVII)
27. The Role of Triangular Numbers in the Pellian Equation x2 −Dy2 = 1 (Part XVIII)
28. Consecutive Odd Numbers and the Pellian Equation (Part XIXA)
29. Consecutive Odd Numbers and the Pellian Equation (Part XIXB)
30. Triangular and the Odd Numbers and the Pellian Equation (Part XIXC)
31. Triangular Numbers and the Pellian Equation x2 −Dy2 = 1 (Part XX)
32. The Pellian Equation x2 −Dy2 = ±1 from Dual Sequences (Part XXI)
  
Table of Contents A2
  
New Methods-Loubère & Méziriac Section
  
  1. General Method for Staircase Squares (Part I)
  2. General Method for Large Order Staircase Squares (Part II)
  3. Siamese and Uniform Step Squares (Part III)
  4. Siamese and Uniform Step Squares - Analogy to Fractals (Part III Con't)
  5. Computer Algorithms for Staircase and Uniform Step Squares (Part IV)
  6. Computer Algorithms for Staircase and Uniform Step Squares (Part V)
  7. A Sequence for Generating Siamese and Uniform Step SquareTypes (Part VI)
  
  8. New Method for Méziriac Type Squares (Part IA)
  9. New Method for Méziriac Type Squares (Part IB)
  10. New Method for Méziriac Type Squares (Part II)
  11. New Method for Méziriac Type Squares (Part III)
  12. New Method for Méziriac Type Squares (Part IV)
 13. New Method & Rules for Loubère Type Squares (Part I)
 14. New Method & Rules for Loubère Type Squares (Part II)
 15. New Method & Rules for Left Diagonal Staircase Squares (Part I)
 16. Novel 11th Order Staircase Squares (Part II)
 17. Magic Squares Wheel Method-Redux (Part I)
 18. Magic Squares Wheel Method-Redux (Part II)
 19. Magic Squares Wheel Method-Redux (Part III)
 20. Magic Squares Wheel Method-Redux (Part IV)
 21. Magic Squares Wheel Method-Redux (Part V)
 22. Magic Squares Wheel Method-Redux (Part VI)
 23. New Method for Odd Magic Squares
  
Page 1 Square of Squares - Imaginary Numbers in Triples Table of Contents A3
  
Page 1 Wheel Section Table of Contents A4
  
Page 2 Interleaved Sequences and Bremner Type Squares Table of Contents II
  
Table of Contents A3
  
Imaginary and Square of Squares Section
  
1. Use of Imaginary Numbers for Square of Squares (Part IA)
2. Table of Tuples Using Imaginary Numbers for Square of Squares (Part IIA)
3. Table of Tuples and Use of Magic Ratio (R) for Tuple Conversion (Part IB)
4. Further Use of the Magic Ratio (R) and Relation to the Golden Mean
5. Table of Tuples for Square of Squares (Part IC)
6. Table of Tuples for Square of Squares (Part IIC)
7. Table of Tuples for Square of Squares (Part IIIC)
8. Table of Tuples for Square of Squares (Part ID)
9. Table of Tuples for Square of Squares (Part IE)
10. Conversion of Tuples to Higher Series of Tuples (Part IF)
11. Conversion of Tuples to Higher Series of Tuples (Part IIF)
12. Conversion of Tuples to Higher Series of Tuples (Part IIIF)
13. Geometric Progression of Tuple Numbers to Generate New Sequences (Part IVF)
14. Geometric Progression and Recursive Methods to Generate New Sequences (Part VF)
15. Recursion Methods to Generate New Integer Sequences (Part VIF)
16. Squares of Seven Squares using Imaginary Numbers (Part IG)
Table of Contents A4
  
New Wheel/Octagon Section 2022
  
1. Octagon Algorithm for Wheel Magic Squares (Part I)
2. Octagon Algorithm for Wheel Magic Squares (Part II)
3. Octagon Algorithm for Wheel Magic Squares (Part III)
4. Octagon Algorithm for Border Wheel Squares (Part IV)
5. Octagon Algorithm for Border Wheel Squares (Part V)
6. Octagon Algorithm for Border Wheel Squares (Part VI)
7. Octagon Algorithm for Border Wheel Squares (Part VII)
8. Border Wheel Squares-Number of Squares (Part VIIIa)
9. The Wheel Octagon Triangle - Ascending Diagonals (Part VIIIb)
10. The Even/Odd Triangles - Ascending Diagonals (Part VIIIc)
11. Octagon Algorithm for Border Wheel Squares (Part IX)
12. Octagon Algorithm for Border Wheel Squares (Part X)
13. Octagon Algorithm for Border Wheel Squares (Part XI)
14. Octagon Algorithm for Border Wheel Squares (Part XII)
15. Octagon Algorithm for Border Wheel Squares (Part XIII)
16. Octagon Algorithm for Border Wheel Squares (Part XIV)
17. Octagon Algorithm for Border Wheel Squares (Part XV)
18. Octagon Algorithm for Border Wheel Squares (Part XVI)
19. Octagon Algorithm for Border Wheel Squares (Part XVII)
  
Table of Contents A5
  
Wheel Section
  
1. Introduction and Discussion
2. Magic Square Wheel Method A-1:Variant 1
3. Magic Square Wheel Method A-1:Variant 2
4. Magic Square Wheel Method A-1:Variant 2 7x7
5. Magic Square Wheel Method A-1:a 9x9 Variant
6. Magic Square Wheel Method A-2 Template Inversion
7. Method B Wheel Expansion
8. Method C: Expansion of Wheel Squares (Part IIIB)
9. Method C: Continuation of (Part IIIB)
10. Method D: Wheel Spoke/Anti-Spoke Method (Part IV)
11. New 7x7 Magic Squares Wheel Method (Part I)
12. New 7x7 Magic Squares Wheel Method (Part II)
13. New 9x9 Magic Squares Wheel Method (Part III)
14. New 9x9 Magic Squares Wheel Method (Part IV)
15. New 9x9 Magic Squares Wheel Method (Part V)
16. New 7x7 Magic Squares Wheel Method (Part VI)
17. New 9x9 Magic Squares Wheel Method (Part VII)
18. New 7x7 Magic Squares Wheel Method (Part VIII)
19. New 9x9 Magic Squares Wheel Method (Part IX)
20. New 9x9 Border Magic Squares Wheel Method (Part X1)
21. New 9x9 Border Magic Squares Wheel Method (Part X2)
22. New 11x11 Border Magic Squares Wheel Method (Part X3)
23. New 11x11 Border Magic Squares Wheel Method (Part X4)

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