Numbers
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1  Diophantine Equations
1.1  The 450 Pound Problem (x^3 + y^3 = 6z^3)	5
1.2  On  x^2 + y^3 = z^6	7
1.3  On x^3 – x + y^3 – y = z^3 – z	9
1.4  Sums of Consecutive Nth Powers Equal to Nth Power	14
1.5  Concordant Forms	21
1.6  N = (x^2 + y^2)/(1+xy) is a Square	26
1.7  If ab+1, ac+1, bc+1 are Squares	28
1.8  Sums of Three Cubes	38
1.9  Numbers Expressible As (a^2 – 1)(b^2 – 1)	40
1.10  Bi-Rational Substitutions Giving Squares	45
1.11  Miscellaneous Diophantine Equations	47
1.12  Triangles and Diophantine n-tuples	51
1.13  Diophantine n-tuples and their Duals	60
2  On Fermat
2.1  Generalized Little Theorem of Fermat	70
2.2  Barlow's Observation	71
2.3  Sums of Powers in Terms of Symmetric Functions	73
2.4  On Case 1 of Fermat's Last Theorem	78
2.5  Why z Is Not a Prime Power in z^p = x^p + y^p	85
2.6  Fermat's Infinite Descent	87
2.7  Fermat's Last Theorem for Cubes	88
2.8  Kummer's Objection	94
2.9  Fermat's Last Theorem for Quadratic Integers	95
3  Problems
3.1  Diophantine Walk-a-thon	98
3.2  Geodesic Diophantine Boxes	99
3.3  The Two Ohm Problem	107
3.4  One In The Chamber	113
3.5  Square Triangular Numbers	115
3.6  No Equilateral Triangles on a Chess Board	117
3.7  Center of Gravity With Integer Coordinates	118
3.8  Smallest Quad With Integer Sides, Perp Diags	121
3.9  Tetrahedra with Edges in Arithmetic Progression	124
3.10  Double Equations from Triangles in Squares	131
3.11  Clouds, Shy Squares, and Diophantus	139
3.12  No Four Squares In Arithmetic Progression	141
4  Linear Recurring Sequences
4.1  Identities for Linear Recurring Sequences	150
4.2  Pseudoprimes For x^2 – 4x – 9	151
4.3  Continued Fractions and Characteristic Recurrences	151
4.4  Recurrences and Pell Equations	153
4.5  Some Properties of the Lucas Sequence	156
4.6  Periods of Fibonacci Sequences Mod m	165
4.7  Summations and Recurrences	167
4.8  Perrin's Sequence	169
5  Sequences
5.1  Fibonacci, 1/89, And All That	174
5.2  Sequence Partitionable Into Powers of 2 or 3	176
5.3  Integer Sequences Related To π	178
5.4  Anti-Symmetric Arrays for Linear Recurrences	182
5.5  Sequences With No Arithmetic Progressions	185
6  Unit Fractions
6.1  Unit Fraction Partitions	187
6.2  The Greedy Algorithm for Unit Fractions	191
6.3  Odd Greedy Unit Fraction Expansions	197
6.4  Reverse Greed for Unit Fractions	201
6.5  Minimizing Denominators of Unit Fraction Expansions	204
6.6  Unit Fractions and Fibonacci	209
7  Magic Squares
7.1  Solving Magic Squares	211
7.2  Magic Square of Squares	216
7.3  Orthomagic Square of Squares	223
7.4  Automedian Triangles, Eigen Vectors, Magic Squares	226
7.5  Discordance Impedes Square Magic	232
7.6  No Progression of Four Rectangles On A Conic?	239
8  Digits
8.1  Sum-of-Digits Iterations	246
8.2  Geometric Dot Products and Digit Reversals	248
8.3  Reverse Digit Sums Leading to Palindromes	250
8.4  Self-Similar Reverse-Sum Sequences	256
8.5  On General Palindromic Numbers	265
8.6  Least Significant Non-Zero Digit of n!	269
8.7  The Factorial Number System	273
8.8  Catch of the Day (153 Fishes)	274
8.9  Infinitely Many Rhondas	275
8.10  Is e Normal?	277
8.11  In Defense of Base-Related Problems	285
9  Congruences
9.1  A Knot of Congruences	288
9.2  Limit Cycles of xy (mod x+y)	289
9.3  More Results on the Form xy (mod x+y)	293
9.4  Quadratic Congruences	297
9.5  Squares in Arithmetic Progression (mod p)	298
10  Primes
10.1  Reflective and Cyclic Sets of Primes	304
10.2  On the Density of Some Exceptional Primes	307
10.3  Highly Wilsonian Primes	311
10.4  Is This a Prime?	313
10.5  Legendre's Prime Number Conjecture	314
10.6  A Primality Criterion	316
10.7  Lucas's Primality Test With Factored N-1	317
11  Divisibility
11.1  Cyclic Divisibility	319
11.2  Cyclic and Reverse Divisibility	322
11.3  Product Divisible By Sum of Squares	328
11.4  Divisors of an n-term Geometric Series	331
11.5  Factoring Zeta	332
11.6  The Euclidean Algorithm	335
11.7  Irreducibility Criteria	337
11.8  Detecting Squares	340
11.9  Can n Divide !n ?	341
11.10  Powers of Primes Dividing Factorials	344
12  Arithmetic Functions
12.1  Sum of Divisors Equals a Power of 2	347
12.2  The Half-Totient Tree	348
12.3  Main Diagonals and Euler's Totient	352
12.4  The Distribution of Perfection	353
12.5  Differently Perfect	354
12.6  Coprime Partitions	356
12.7  Average of σ(n)/n	358
12.8  Congruences Involving the Totient Function	361
12.9  Sublime Numbers	365
13  Sum of prime factors
13.1  Definitions and Basic Propositions	368
13.2  Primes Related to Corollary 7.1	377
13.3  Iterated Functions	383
13.4  Higher-Order Solutions	386
13.5  Miscellaneous Related Results	396
13.6  Sum of Prime Factors of Linear Polynomials	403
13.7  Sum of Prime Factors of Quadratic Polynomials	406
14  Miscellaneous
14.1  Rounding Up To π	410
14.2  The Dartboard Sequence	414
14.3  Numeri Idonei	416
14.4  Evidence for Goldbach	419
14.5  Four Squares From Three Numbers	422
14.6  Anti-Carmichael Pairs	425
14.7  Zeisel Numbers	426
14.8  Coherent Arrays of Squares	427
14.9  Infinite Descent versus Induction	430
14.10  A Special Property of 151	436
14.11  Differences Between Powers	439
14.12  Exponential Partitions	445
14.13  Primitive Roots and Exponential Iterations	449
14.14  Quadratic Reciprocity – The Jewel of Arithmetic	456
15  Symmetric Pseudoprimes
15.1  Introduction	461
15.2  Basic Propositions
15.2.1  Nomenclature and Definitions	462
15.2.2  Basis Sequences	463
15.2.3  Elementary Symmetric Functions	465
15.2.4  Reversible Sequences	467
15.2.5  Coefficient Sequences	469
15.2.6  Relation Between Basis/Symmetric Functions	470
15.2.7  Primes Types Relative to f(x)	472
15.2.8  Recurrences Modulo Composites	475
15.3  Pseudoprimes Relative to Selected Polynomials
15.3.1  Linear	476
15.3.2  Quadratic	477
15.3.3  Cubic	482
15.3.4  Quartic	485
15.3.5  Quintic	487
15.4  Congruence Conditions on Linear Recurring Sequences
15.4.1  Second Order	492
15.4.2  Third Order	493
15.4.3  Fourth Order	495
Appendix A:  Sums of Prime Factors	505
Appendix B:  Symmetric Pseudoprimes	519
Appendix C:  Concordant Forms	526
Appendix D:  Table of Geodesic Boxes	529
Appendix E:  Odd-Greedy Expansion of 5/139	536