PrimeProducing Polynomials Diophantine Polynomials

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Prime-Producing Polynomials(Diophantine
Polynomials)

• By Mike Munroe
• (Senior in Mechanical Engineering)

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Special Thanks

• Prof. Adam Avilez, my advisor
• My Beautiful Talented Wife, Micah
• Gerin The CFD Lab gurus at ASU

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Table of Contents

• What is a Race?
• Historic Polynomials
• Our Race
• 100 by 100 plot
• Its not noise
• Filtering shows structure
• In 3-d
• Paths in MatLab

• Paths in Excel
• Fitting the Curve
• Depth Richness
• Paths of Interest
• Comparing Paths
• Reparametrized Field
• Next Steps
• Questions
• Extras?

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What is a Prime-Producing Polynomial Race?

• Given A Polynomial ax2 bx c
• and x 0,1, 2, . . . , n
• Find Integer values of a, b, and c so the
  polynomial exhibits a prime rich range for the
  given domain.
• Richness Number of Primes
• n1

The one with the richest polynomial wins.
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Famous Prime-Producing Polynomials

• Euler (1772)
• x2 x 41
• for x 0-39 ? 100
• for x 0-100 ? 87
• for x 0-255 ? 73
• Legendre (1798)
• x2 x 17
• for x 0-15 ? 100
• for x 0-100 ? 60
• for x 0-255 ? 47

• Fung Ruby (1987?)
• 36x2 810x 2753
• for x 0-44 ? 100
• for x 0-100 ? 68
• for x 0-255 ? 60

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THIS Prime-Producing Polynomial Race

• Given A Polynomial ax2 bx c
• and x 0,1, 2, . . . , 255
• Find Values of a, b, and c producing the most
  prime numbers
• Richness Number of Primes 50
• n

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100 X 100 Contour
Governing Equation

• Each color
• is a richness interval
• is interpolated
• Shows a level surface
• Patterns are hard
• to detect.
• Mean Richness 11.47
• St. Dev Richness 2.67

Richness
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Larger View
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Comparison to Noise
NOTICE These Two Are Not Similar!! Is this
surprising?
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100 X 100 Faceted Plot
Governing Equation

• Each integer point
• represents 1 quadratic
• has 1 color
• Patterns are harder
• to detect.

Richness
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Three Level Surfaces
Governing Equation

• Observations
• Vertical line trends
• Still Noisy

Richness
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Larger View
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Filter 30 Rich
Governing Equation

• Each color
• is a richness interval
• is interpolated
• Shows a level surface
• (20 surfaces default)
• Clearer
• Still contains noise

Richness
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Filtered AND Three Levels
Filter above 45 rich 3 level curves
Note Parabolic paths Clear sky Richer peaks
have a larger footprint
Richness
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3-D image
Number of primes produced
c axis
b - axis
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Paths in MatLab
c - axis
b - axis
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Path in Excel
c - axis
b - axis
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Fit curve of paths
In excel these path fit a parabola to round-off
error for machine!
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General Equation of path

• General Space

• Form of Path in space

• Plane a1 in Space

• Path on Plane

• where

• is constant for each path and will be called the
  intercept for the path.

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Depth of path

• Path of interest
• a1, c141
• Eulers Path
• Observations
• Richer near 0
• Stabler near 0
• Mean near 43
• Poorer toward 8
• Unstabler near 8

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Richness of path
Depth into Eulers Path
Any richness under 50 was grounded to
zero The first one occurred at step
1088. Subsequent groundings occurred
intermittently
The prime-rich polynomial at step 1088 is
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Paths to explore
General Equation for path B0.25(C)2(intercept-0
.25)

• Lesson Learned
• Look along path
• Look near the C-axis

Paths of interest B0.25(C)2(41-0.25) B0.25(C)2
(19421 -0.25) B0.25(C)2(21377-0.25) B0.25(C)2
(55661 -0.25) B0.25(C)2(115721 -0.25)

• Some Rich Path-Intercepts
• 41 -gt74.1
• 19421 -gt70.2
• 21377 -gt72.9
• 55661 -gt73.3
• 115721 -gt70.5

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Comparing Paths
Higher than intercept
Plateaus
A few steps in
Many Steps in
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Parametrized field
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Investigate Reparametrization
Why the variation? Method to filter? Relationshi
p of richness descent to intercept
location? Relationship of richness descent to
intercept richness?
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Unanswered Questions

• Patterns
• Vertical (AC Gap Frequency)
• Diagonal (Parabolic Sheets)
• A-axis phenomenology
• Depth characterization (along polynomial)
• Symmetry
• Numbers within the polynomials
• The BIG search

• Line Coefficient richness vs. Line Polynomial
  richness
• Strong pseudoprime paths

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Vertical / Front View
What causes these gap-patterns?
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The B1 Plane
Unfiltered
Filtered
These are parabolic sheets in the polynomial
space
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Language options
Matlab 699 Billion years or 46.6 (15e9)
yrs C 3.26 days Ratio C(3 x 107 )Matlab
Conclusion Use C, or possibly Fortran for big
jobs
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Any Questions?
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Resonance
Fourier Transform of the data reveals resonances
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The C1 plane
Unfiltered
Filtered
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Larger 3d space
Compute the richness for A, B, C 1 to
100 Question What is the richness of the
richest polynomial in this space? Answer There
are three polynomials with richness of
0.741176470588235 Is that likely?
What does this mean?
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Larger 3d-scatter
Note -parabolic paths -rich at a2 -multiple
a-planes
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Horizontal / Side View
There are patterns on both axes