Appendix D List of Figures
1 Prologue
Figure 1.4.1 FoxTrot comic
3 From Linear Equations to Geometry
Figure 3.2.1 Solutions to a linear Diophantine equation
Figure 3.3.2 Positive solutions to a linear Diophantine equation
Figure 3.5.1 Visualizing when a cube is one less than a square
6 Prime Time
Figure 6.2.2 Part of Euclid IX.20 proof
7 First Steps With General Congruences
Figure 7.6.2 Solutions of a typical Mordell curve
Figure 7.8.1 Cutting a cake with \(7\) candles using two cuts
Figure 7.8.2 Cutting a cake with \(6\) candles using two cuts
Figure 7.8.4 Stellated \(7\)-gons
8 The Group of Integers Modulo \(n\)
Figure 8.1.2 Addition table for \(\mathbb{Z}_3\)
Figure 8.1.3 Multiplication table for \(\mathbb{Z}_3\)
Figure 8.1.4 Visualizing multiplication modulo \(n=7\)
Figure 8.2.1 Visualizing powers modulo \(n=11\)
10 Primitive Roots
Figure 10.0.1 Visualizing powers modulo \(n=11\) (again)
Figure 10.1.2 Visualizing powers modulo \(n=10\)
12 Some Theory Behind Cryptography
Figure 12.2.1 Visualizing powers modulo \(n=11\) (yet again)
Figure 12.3.1 Visualizing powers modulo \(n=11\) (last time)
13 Sums of Squares
Figure 13.1.5 Five as a sum of squares
Figure 13.2.5 Thirteen as a sum of squares
Figure 13.4.2 A different lattice for finding sums of squares
Figure 13.4.7 Adding circles to the sum of squares helper lattice
Figure 13.4.8 Sum of squares helper lattice with triangles and circles
Figure 13.4.14 Sum of squares helper lattice with all the bells and whistles
14 Beyond Sums of Squares
Figure 14.1.4 Factoring in the Gaussian integers
Figure 14.1.6 Plot of Gaussian primes
15 Points on Curves
Figure 15.0.1 Integer points on ellipses
Figure 15.1.1 Rational points on a circle
Figure 15.1.7 Rational points on an ellipse
Figure 15.2.2 Rational points on Dudeney’s curve
Figure 15.4.1 Integer points on an ellipse
Figure 15.4.5 Integer points on a parabola
Figure 15.4.6 Finding more integer points on a parabola
Figure 15.5.3 Finding more integer points on an ellipse
Figure 15.5.4 Integer points on a hyperbola
Figure 15.5.5 Finding more integer points on a hyperbola
16 Solving Quadratic Congruences
Figure 16.3.5 Lagrange’s Table III of divisors of certain integers
Figure 16.3.6 Lagrange’s Table IV of divisors of certain integers
Figure 16.4.10 Visualizing powers modulo \(n=13\)
Figure 16.5.1 Visualizing powers modulo \(n=13\) (again)
17 Quadratic Reciprocity
Figure 17.6.1 Geometric interpretation of power in Eisenstein criterion
Figure 17.6.4 Visualizing the proof of quadratic reciprocity
18 An Introduction to Functions
Figure 18.1.3 Plotting \(\phi\)
Figure 18.2.7 Visualizing sums of squares as area
19 Counting and Summing Divisors
Figure 19.4.12 Mersenne and amicable numbers
Figure 19.6.1 Mersenne and \(3\)-perfect numbers
20 Long-Term Function Behavior
Figure 20.1.1 Error bounds for the sum of squares
Figure 20.1.4 A better error bound for the sum of squares
Figure 20.2.1 The average value of \(\tau\)
Figure 20.2.3 The average value of \(\tau\) out a long ways
Figure 20.3.1 Visualizing \(\tau\) as lattice points
Figure 20.3.2 Visualizing lattice points as area for \(\tau\)
Figure 20.3.4 Visualizing the error in areas for \(\tau\)
Figure 20.3.8 More symmetry and the average of \(\tau\)
Figure 20.3.9 Visualizing \(\gamma\)
Figure 20.3.12 An even more precise error estimate for the average of \(\tau\)
Figure 20.4.1 Lattice points and \(\sigma\)
Figure 20.4.3 The average value of \(\sigma\)
Figure 20.4.4 The average value of \(\sigma\) compared with a straight line
21 The Prime Counting Function
Figure 21.1.1 A first plot of prime \(\pi\)
Figure 21.1.6 Comparing primes to the log of log
Figure 21.2.3 Comparing primes to the log integral
Figure 21.2.4 Excerpt from a letter of Gauss about primes
Figure 21.2.5 Comparing prime counting and log integral again
Figure 21.3.8 Comparing prime counting to \(2x/\log(x)\)
Figure 21.4.1 The prime counting function \(\pi(x)\text{,}\) again
Figure 21.4.2 The Chebyshev theta function
Figure 21.4.5 Limiting values of Chebyshev and others
Figure 21.4.8 A new way to envision the prime \(\pi\) function
Figure 21.4.9 Integrals of \(1/\log(x)\)
22 More on Prime Numbers
Figure 22.1.7 The modulo \(4\) prime race
Figure 22.1.8 The modulo \(8\) prime race
Figure 22.3.5 Estimating twin primes
24 Infinite Sums and Products
Figure 24.2.2 The Riemann zeta function
Figure 24.6.1 Integer lattice points visible from the origin
Figure 24.6.5 The average value of \(\phi\)
Figure 24.6.6 Labeled lattice points for \(\phi\)
25 Further Up and Further In
Figure 25.1.1 Comparing prime \(\pi\) with another estimate
Figure 25.2.3 Von Koch error estimate for prime \(\pi\)
Figure 25.3.1 The Riemann zeta function again
Figure 25.3.2 Visualizing the Riemann zeta as a complex function
Figure 25.3.3 Visualizing the Riemann zeta in three dimensions
Figure 25.3.4 Comparing complex and real plots of Riemann zeta
Figure 25.3.6 Riemann zeta along a specific line
Figure 25.4.1 The \(J\) function
Figure 25.5.1 A better approximation to prime \(\pi\text{!}\)
Figure 25.6.1 Plotting the log integral along a line in \(\mathbb{C}\)
Figure 25.7.2 The better approximation to prime \(\pi\text{,}\) again
Figure 25.8.3 A case of the Sato-Tate conjecture
