Computer discovery and analysis of large Poisson polynomials
David H. Bailey
http://www.davidhbailey.com
Lawrence Berkeley National Lab (retired)
University of California, Davis, Dept. of Computer Science
Collaborators:
Jonathan M. Borwein (deceased 2 Aug 2016), University of Newcastle, Australia
Jason Kimberley, University of Newcastle, Australia
Watson Ladd, University of California, Berkeley

October 3, 2017
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Jonathan M. Borwein, 1951–2016
I 388 published journal articles; another 103 inrefereed conference proceedings.
I ISI Web of Knowledge lists 6,593 citations from351 items; one paper has been cited 666 times.
I His work spanned pure mathematics, appliedmathematics, optimization theory, computerscience, mathematical finance, and experimental
mathematics.
I Borwein sought to do research that is accessible,

and to highlight aspects of his work that a broad
audience (including both researchers and the lay
public) could appreciate.
I More information, including memorials and links tonearly 1700 publications, preprints and talks:
http://www.jonborwein.org.

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Standing on the shoulders of giants
The following study is a paradigm of multidisciplinary experimental mathematics, as it
crucially relies on many highly talented contributions:
I Work by Brent, Zimmermann, Lefevre and other developers of the MPFR

package, which was used for extreme precision computation.
I An enormous software infrastructure behind our computer code:

GNU compilers and Apple’s Berkeley Unix software.
Fortran custom datatypes and operator overloading.
I OpenMP software for parallel processing.
I
I

I Ferguson’s PSLQ algorithm (as far as we are aware, this study involves the largest

computations ever done using PSLQ).
I Crandall’s work applying the Poisson equation to image enhancement.
I Jon Bowein’s derivation of a much more rapidly convergent algorithm for the
Poisson phi function.
I Numerous studies involving elliptic curves, theta functions, ideals and fields.
I A key observation by Jason Kimberley of the University of Newcastle, Australia.
I A concluding proof by Watson Ladd, a graduate student at U.C. Berkeley.
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The PSLQ integer relation algorithm
Given a vector (xn ) of real numbers, an integer relation algorithm finds integers (an )such thata1 x1 + a2 x2 + · · · + an xn = 0(to within the precision of the arithmetic being used), or else finds bounds within
which no relation can exist.
Helaman Ferguson’s PSLQ algorithm is the most widely used integer relation
algorithm, although variants of the LLL algorithm can also be used.
Integer relation detection (using PSLQ or any other algorithm) requires very highnumeric precision, both in the input data and in the operation of the algorithm.
1. H. R. P. Ferguson, D. H. Bailey and S. Arno, “Analysis of PSLQ, an integer relation finding
algorithm,” Mathematics of Computation, vol. 68, no. 225 (Jan 1999), 351–369.
2. D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques andapplications,” Mathematics of Computation, vol. 70, no. 236 (Oct 2000), 1719–1736.
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Helaman Ferguson’s “Umbilic Torus SC” sculpture at Stony Brook Univ.

5 / 34

PSLQ, continued
I PSLQ constructs a sequence of integer-valued matrices Bn that reduce the vectory = x · Bn , until either the relation is found (as one of the columns of matrix Bn ),or else precision is exhausted.
I A relation is detected when the size of smallest entry of the y vector suddenly
drops to roughly “epsilon” (i.e. 10−p , where p precision in digits).
I The size of this drop can be viewed as a “confidence level” that the relation is nota numerical artifact: a drop of 20+ orders of magnitude almost always indicates areal relation.
Efficient variants of PSLQ:
I 2-level and 3-level PSLQ perform almost all iterations with only double precision,

updating full-precision arrays as needed. They are hundreds of times faster thanthe original PSLQ.
I Multi-pair PSLQ dramatically reduces the number of iterations required. It was

designed for parallel systems, but runs faster even on 1 CPU.
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Decrease of log 10(min |yi |) in multipair PSLQ run

7 / 34

Application of multipair PSLQ
One simple but important application of multipair PSLQ is to recognize a computed
numerical value as the root of an integer polynomial of degree m.
Example: The following constant is suspected to be an algebraic number:
α = 1.232688913061443445331472869611255647068988824547930576057634684778 . . .
What is its minimal polynomial?
Method: Compute the vector (1, α, α2 , · · · , αm ) for m = 30, then input this vector tomultipair PSLQ.
Answer (using 250-digit arithmetic):
0 = 697 − 1440α − 20520α2 − 98280α3 − 102060α4 − 1458α5 + 80α6 − 43920α7+ 538380α8 − 336420α9 + 1215α10 − 80α12 − 56160α13 − 135540α14 − 540α15+ 40α18 − 7380α19 + 135α20 − 10α24 − 18α25 + α30
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The Poisson potential function
In 2012, Richard Crandall, while investigating techniques to sharpen images, notedthat each pixel was given by a form of the 2-D Poisson potential function:
1 X cos(mπx ) cos(nπy )φ2 (x , y ) = 2
π m,n odd
m2 + n2
In a 2013 study, we numerically discovered, and then proved the intriguing fact that for
rational (x , y ),1
φ2 (x , y ) = log απwhere α is algebraic, i.e., the root of a some integer polynomial of degree m.
By computing high-precision numerical values of φ2 (x , y ) for various specific rational x
and y , and applying a multipair PSLQ program, we were able to produce the explicit
minimal polynomials for α in numerous specific cases.
I D. H. Bailey, J. M. Borwein, R. E. Crandall and J. Zucker, “Lattice sums arising from the Poisson

equation,” Journal of Physics A: Mathematical and Theoretical, vol. 46 (2013), 115201.

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Samples of minimal polynomials found by multipair PSLQ
s
5
6
7
8
9

10

Minimal polynomial corresponding to x = y = 1/s:
1 + 52α − 26α2 − 12α3 + α4
1 − 28α + 6α2 − 28α3 + α4
−1 − 196α + 1302α2 − 14756α3 + 15673α4 + 42168α5 − 111916α6 + 82264α7−35231α8 + 19852α9 − 2954α10 − 308α11 + 7α12
1 − 88α + 92α2 − 872α3 + 1990α4 − 872α5 + 92α6 − 88α7 + α8
−1 − 534α + 10923α2 − 342864α3 + 2304684α4 − 7820712α5 + 13729068α6−22321584α7 + 39775986α8 − 44431044α9 + 19899882α10 + 3546576α11−8458020α12 + 4009176α13 − 273348α14 + 121392α15−11385α16 − 342α17 + 3α18
1 − 216α + 860α2 − 744α3 + 454α4 − 744α5 + 860α6 − 216α7 + α8These computations are very expensive. The case x = y = 1/32, for instance, required
10,000-digit arithmetic and ran for 45 hours. Other runs, using even higher precision,
ultimately failed, evidently due to subtle program bugs. Help!

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Kimberley’s formula for the degree of the polynomial
Based on our preliminary results, Jason Kimberley of the University of Newcastle,
Australia observed that the degree m(s) of the minimal polynomial associated with thecase x = y = 1/s appears to be given by the following:
Set m(2) = 1/2. Otherwise for primes p congruent to 1 mod 4, set m(p) = int2 (p/2),
where int denotes greatest integer, and for primes p congruent to 3 mod 4, setm(p) = int (p/2)(int (p/2) + 1). Then for any other positive integer s whose prime
factorization is s = p1e1 p2e2 · · · prer ,?

r −1m(s) = 4rY
2(ei −1)pim(pi ).

i=1

Does Kimberley’s formula hold for larger s? Why?
What is the true mathematical connection between the pair of rationals (x , y ) and thealgebraic number α?
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Three improvements to the Poisson polynomial computation program
1. MPFUN2015: A new thread-safe multiprecision package.
I

Speedup: 3X

2. A new 3-level multipair PSLQ program.
I

Speedup: 4.2X

3. Parallel implementation on a 16-core system.
I

Speedup: 12.2X

Overall speedup: 156X

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Thread safety in high-precision software
Even though parallel implementations are often required for high-precisioncomputations, most high-precision software packages are not thread-safe and thus
cannot be used in shared memory parallel programs.
I Many packages employ global read/write variables, e.g., for transcendental

function evaluation, which ruin thread safety.
I The working precision level, a global variable that is changed frequently within the

package itself and often by users also, is particularly troublesome.
One bright spot: the MPFR package
I Thread-safe (if compiled with the thread-safe option).
I Correct rounding to the last bit.
I Fastest package currently available.
I Based on the Gnu multiprecision package (GMP).
I Low-level arithmetic and transcendental functions only.
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MPFUN2015: DHB’s thread-safe arbitrary precision package
Available in two versions:I MPFUN-Fort: Completely self-contained, all-Fortran version. Compilation is a

simple one-line command, which completes in a few seconds.
I MPFUN-MPFR: Calls the MPFR package for lower-level operations. Installation

is significantly more complicated (since GMP and MPFR must first be installed),
but performance is roughly 3X faster. We used MPFUN-MPFR in this study.
Both versions include a high-level language interface, using custom datatypes andoperator overloading — for most applications, only a few minor changes to existing
double precision code are required.
Designed for high-level Fortran programs; a C++ version is planned but not written.
I D. H. Bailey, “MPFUN2015: A thread-safe arbitrary precision computation package,”manuscript, http://www.davidhbailey.com/dhbpapers/mpfun2015.pdf.
I Software is available at http://www.davidhbailey.com/dhbsoftware.
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New three-level multipair PSLQ program
Employs three levels of numeric precision:
I Ordinary double precision.
I Medium precision, typically 100–2000 digits.
I Full precision, typically many thousands of digits.

When an entry of the double precision reduced vector is smaller than 10−14 , themedium precision arrays are updated by matrix multiplication.
Similarly, when an entry of the medium precision reduced vector is smaller than themedium precision “epsilon,” the full-precision arrays are updated by matrix
multiplication.
Substantial care must be taken to manage this three-level hierarchy, and to correctly
handle numerous atypical scenarios.

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Jon Borwein’s fast algorithm to compute φ2 (x , y )φ2 (x , y ) =1
θ2 (z, q)θ4 (z, q)log,
2π
θ1 (z, q)θ3 (z, q)

where q = e −π and z = π2 (y + ix ). Compute the four theta functions using thefollowing very rapidly convergent formulas involving complex variables:θ1 (z, q) = 2
θ2 (z, q) = 2∞
X2(−1)k−1 q (2k−1) /4 sin((2k − 1)z),

k=1
∞
X

2

q (2k−1) /4 cos((2k − 1)z),k=1

θ3 (z, q) = 1 + 2
θ4 (z, q) = 1 + 2

∞
X
k=1
∞
X

2

q k cos(2kz),
2(−1)k q k cos(2kz).

k=1
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High-level computational algorithm
1. Given rationals x = p/s and y = q/s, select a conjectured minimal polynomial
degree m(s) (using Kimberley’s formula) and other parameters for the run.
2. Calculate φ2 (x , y ) to P2 -digit precision using the formulas from two viewgraphs
above. When done, calculate α = exp(8πφ2 (x , y )) and generate the (m + 1)-longvector X = (1, α, α2 , · · · , αm ), to P2 -digit precision.
3. Apply the three-level multipair PSLQ algorithm to X . For larger problems, employ
a parallel version of the three-level multipair PSLQ code, using the OpenMP
construct DO PARALLEL to perform certain time-intensive loops in parallel.
4. If no numerically significant relation is found, try again with a larger degree m or
higher precision P2 . If a relation is found, employ the polynomial factorization
facilities in Mathematica and Maple to ensure that the polynomial is irreducible.

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Application program and libraries for the Poisson calculations
Description
Poisson polynomial program∗
MPFUN-MPFR package
MPFR package
GMP package
Total

Language
FortranFortranCC

Lines of code
2,000
12,00093,00083,000190,000∗ This includes the computation of φ (x , y ) and the 3-level multipair PSLQ program.
2

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192-degree minimal polynomial found by multipair PSLQ for x = y = 1/35
−1−127936α1−119729568α2+86259615424α3−63414593420752α4−23761356230294720α5−21521006149475920864α6+611939502140287079872α7−368241444434585386649704α8+34534137997535622208800832α9+190487328667825240861480928α10+70731866698900460686569086656α11−30639321248717091789168538879600α12+1922209242084456061370070376570688α13−92225937246234731111918315818791008α14+5237085972458845348795423777776516544α15−235187168740813424983476512954481139732α16+7429223399159195923652315325576416851776α17−183888805409443551177519325744878574722592α18+3271185405201792963719366162166966763587520α19−30299858585212592953207274837044089721625936α20−359992131877582010695076712447360628122541504α21+21102025339314733846086441916632636325491195040α22−476414747523697562008419279360098519444810134848α23+7169682338305684610208768903330082567075749930088α24−77147038334570163492295981291240215533152479746240α25+576164594448741453064385715717563185909190291963744α26−2736845608824484767533372475869861068967680432337984α27+14259726273826198867906144815986191950918103600120720α28−418898418536517746458185180042147268305073686192184768α29+11576727630588055309375415098495616370526328495842993184α30−215864800528660839585002162871715414654518484882221347136α31+2990075428515164844805618172618994337159659201108555443326α32−32103405822691698883721869657536188329165809065032597536320α33+266563559238616545234045835785848715794979563412588549450912α34−1588318769762074127932309766905276050093368042694667151097024α35+4203983024865006644081516487096952711041894782888827508128720α36+45741795572596990194281792252600295928845217432176402329342656α37−888295322846459251495779433538199608179522746791003115766467360α38+9318868074863097497435367536227145321575550333899160360356887616α39−75360568030730663716613169159158032198635749984420852664531432312α40+511904470165868318936871843389229757000581029769096726546021748160α41−3023546278817700158319248501909046229060364677626955289951128897760α42+15762439699463291788277556458704672972787957041647009977980115608384α43−72736254087237112123433235647299134668422519990232924303557659578512α44+294655977143150502457165877012542972911987116693855121083332478262976α45−1023418666617067140855492177628022043641503497647927192890895515496096α46+2872077837131433597858520147550093715024860106120898508958083768560704α47−5244656757384067630568130031166724732476128944651080774800327815250116α48−4375176099824059357126327245955265067024929918868045662842672259365696α49+106846263928061290720995820179791755195031418846948198250527939644422688α50−723639781157123885224054475278025279616831219096501715148796353565656000α51+3859388734169523941501510320228124865152568559427283813598867448036672464α52−18252524309522593867333335673661626604394606878814374964432136569382704704α53+77801314121940709085266334072671847312436136451041737155601244216316832608α54−295924487894698517430363594146438746974018698578472622597771256623091267264α55+990883094386701201740101120896676716687418747972363663196338999871139087736α56−2890647897108739627601089028825831836951153432467718962671967596342224331584α57+7329280206578701233923536747523603867159738577214393279268225554697992024224α58−16529926761499923756105727653427646107756611125773189972311741253023019544512α59+36621401257350141587688985844214102107297318891842289523223570545646432107248α60−97806177157646206090474078202543089426709440278475346399840243186928760712768α61+342578593578669077931375617617876157438332517966938868387799492392630391951328α62−1318494126999138496197980784859753020673610723076044612133782854663456341738176α63+4770648039063802379141989053619421884321933507524681278243277292208295535890193α64−15472388710953109453763060955215809455238864792962040591046439233238251454944640α65+44993126133327412718394996675533962755303579158603905782013384675151509701621696α66−118993678023270686331414029749780909390537125214711043931901528442137124714048640α67+291344068959030996949609862561213601556661749401236700666696596438800158224020448α68−672232544304160020140857460151229781783982323755372371953704546852887742475693952α69+1483756546102154371991621621590988762258290163435879224071206411473630010421819712α70−3160698702072288781771809020419553277725945343183442108817290538424291765729105536α71+6501409535318122589047227869235783252522832570490016457500770280864513874652673648α72−12819483008841743167047686498254296089188806357669061135254105501043843161878338944α73+23940036962388573943673383046497610566473660443396196928270109685668366770350448320α74−41783013070841149166516454269070382975961508352312427230333483156459522180289158272α75+67368007706334464267463678880196649459723255934263643619439168617608299192825564576α76−99642265418236291841000783588241107943576928164973329187609122090423042369846900608α77+135684188062925983721829145899247186881926375911313411751554577363782314702542167104α78−175027359690635875118927274192015444148378662148930449526142186378361018115616958080α79+23061047037097602217994243304224041474746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41879784121657187449503319934776774553449947624311536879983067197120α137+2046994405582919610833914133280166599368341899820908062960382811295175875424α138−475319952147035704834648908024038830280982173827054926876842801020099391040α139+101156139413765995787243855087479738205336119604403640417144416030882137040α140−19536944183496787665897200701914355830359619924729440672204518600726485952α141+3365653573540320310956912862417397989082549665698949749718751351520029216α142−500297582644104844305032452251740515847130656744730715102464280225674048α143+59315729762438733681847579101255419553245420619721207125629069846855484α144−4088968402271371424704252106469344820514085294011337888699549616483264α145−410467411100906673223801525015092827907382852275953565568989408591520α146+244373061239573525170364306606773136690590544856331578133540062357184α147−67137173574208581929099251980225280832815704879993996789961828208784α148+15109453726979262114404719422671376896934991181254480301068783496000α149−3041110741874930638456938010940447684323532404585513230577319188704α150+543474006539496836413675511283771054220572953580018501137140774336α151−82818210860123209370721479679382853454718857015790579069608342392α152+10208909053669116781213384172905303479740054958245095883752727616α153−950607878263673381203873433978784840581837837196127719161731872α154+61537081222427663468738382440653874375219870006734261147376320α155−3614460685557548015719619196028506524834155796764598110372912α156+735146337727640447184879802116259476500819273333226229041984α157−195982957610545601975537049907586449965158403571666750972768α158+35410924134603994553881822516846688340985279903764142254528α159−4586047122379261183418802995335871699189992574849634423682α160+461731528371721725878159689755046846967144056271095432896α161−39548982563052670609179355793571460666409257926095752160α162+3249771246267792941295436011847663684436436914874939968α163−278663520870823657308884684973631236730880421051628656α164+23298555433192575099637973614016576299882162116142016α165−1572480211244215449798675303409379604821388457521312α166+64631732188205970561818215170229051834055602691904α167−359811432336185820581657539275402497742419221400α168−81408350985197555231086181640295965142859955520α169−2670526194482509900914512598125562096574018400α170+824346510881081434287176385644420962607200832α171−40421524575366405147127382177082176049208656α172+859850260877732200189610333732782641323968α173−21746942237477507610278314875487367745056α174+1166204976072113679141136387334878097216α175−16255593447525276321168587153278001172α176−1031047179999272558707644762054830656α177+57315125130135149120405695909830560α178−2190911023714361858443237127579840α179+42156348386945234230569584900496α180−400797347138034992145950828864α181−3239264180104926121034061856α182+57518923520963556655145024α183−302326293499892000458856α184−648961315190292320832α185+4804670128512328736α186+97238301302930240α187−106616047418832α188−478181605696α189+208298592α190+25664α191−α192This distinctive semi-elliptical pattern is strong evidence that the result is correct.

19 / 34

Timings for the case x = y = 1/35
Multiprecision software
ARPRECMPFUN-MPFR
MPFUN-MPFR

PSLQ code
2-level2-level3-level

Cores
1
1
1
2
4
8
16

Run time
1.599 · 106
5.249 · 105
1.240 · 105
7.585 · 104
4.121 · 104
2.476 · 104
1.021 · 104

Speedup
1.00
3.05
12.90
21.08
38.80
64.58
156.61

The run times are wall-clock run times (in seconds), measured on a 16-core 2.4 GHz
MacPro, in a typically busy environment with similar jobs running on other cores.

20 / 34

Selected runs (degrees, precision, timings, etc.) for x = y = 1/s
s
20
24
30
32
34
36
38
40
42
44
46
48
50
52
∗
60
∗
64

m
32
64
64
128
128
144
180
128
192
240
264
256
200
288
256
512

log10 (D)-463.84-1883.78-1868.01-7577.07-7574.93-9570.86-14951.64-7580.00-16993.99-26604.14-32036.34-30248.55-18421.18-38414.49-14477.99-57816.90

P1
160
320
350
650
650
750
900
650
1000
1200
1350
1350
1000
1550
800
1600

P2
700
2200
2300
8200
8200
10300
16000
8200
18000
28000
34000
32000
20000
41000
16000
64000

N
1967
9297
9064
45893
45914
62282
12098445655
15036432376247690241531616894765529190371
802361

M
0.81
11.33
11.33
168.20168.20267.10642.98168.20829.41
2003.33
2921.57
2586.39974.444124.24
336.415172.79

C
1
1
1
1
1
1
1
1
8
8
16
16
8
16
1
16

T (sec.)2.19 · 100
7.73 · 101
1.02 · 102
5.13 · 103
5.16 · 103
9.54 · 103
3.88 · 104
5.02 · 103
1.57 · 104
7.43 · 104
1.06 · 105
8.98 · 104
2.12 · 104
2.12 · 105
5.28 · 103
3.78 · 105

C · T (sec.)
2.19 · 100
7.73 · 101
1.02 · 102
5.13 · 103
5.16 · 103
9.54 · 103
3.88 · 104
5.02 · 103
1.26 · 105
5.94 · 105
1.70 · 106
1.44 · 106
1.69 · 105
3.40 · 106
5.28 · 103
2.42 · 106

s = denominator; m = degree; D = detection level; P1 = medium precision; P2 = full precision; N =number of iterations; M = Mbytes; C = cores; T = wall clock time; C · T = total core-seconds.
21 / 34

Palindromic polynomials
From our results, in the case (1/s, 1/s) where s is even, the resulting polynomial
appears to be palindromic (ak = am−k ). For instance, when s = 16,
p16 (α) = 1 − 1376α1 − 12560α2 − 3550496α3 + 81241720α4 − 169589984α5+ 1334964944α6 − 24307725984α7 + 238934926108α8 − 1043027124704α9+ 2328675366384α10 − 3219896325280α11 + 4238551472456α12− 10247414430048α13 + 28552105805904α14 − 55832851687968α15+ 70020268309062α16− 55832851687968α17 + 28552105805904α18 − 10247414430048α19+ 4238551472456α20 − 3219896325280α21 + 2328675366384α22− 1043027124704α23 + 238934926108α24 − 24307725984α25 + 1334964944α26− 169589984α27 + 81241720α28 − 3550496α29 − 12560α30 − 1376α31 + α32

Nitya Mani, an undergraduate student at Stanford University, observed that if α is aroot of a palindromic polynomial such as this, then α + 1/α is a root of a transformed
polynomial of half the degree. This fact can be used to significantly accelerate thecomputation of Poisson polynomials in the even case (runs denoted by ∗ in the table). 22 / 34

New observations for the case (1/s, 1/s)
After doing some Google searches on the coefficients of the polynomials p11 and p13 ,we found the coefficient 387221579866 in p11 appears in a 2010 preprint by Savin and
Quarfoot, and the coefficient 221753896032 in p13 appears in a manuscript, also dated
2010, by Bostan, Boukraa, Hassani, Maillard, Weil, Zenine and Abarenkova.
Savin and Quarfoot define a sequence ψn of polynomials in x and y , based on the
curve y 2 = x 3 + x , as follows:ψ1 = 1ψ2 = 2yψ3 = 3x 4 + 6x 2 − 1
ψ4 = 2y (2x 6 + 10x 4 − 10x 2 − 2),(1)and, recursively,3
ψ2n+1 = ψn+2 · ψn3 − ψn−1 · ψn+1

for n ≥ 222
ψ2n = 1/(2y ) · ψn (ψn+2 · ψn−1
− ψn−2 · ψn+1)

for n ≥ 3.
23 / 34

Our analysis
We constructeda related sequence Js√of integer coefficient polynomials in a by setting√
x = −a, and so y 2 = x (x 2 + 1) = −a(1 − a); we also remove the leading 2y :J2n+1 (a) = ψ2n+1 (x , y )J2n (a) = 1/(2y ) · ψ2n (x , y )The initial values of Js (a) are
J1 = 1J2 = 1J3 = 3a2 − 6a − 1
J4 = 2a3 − 10a2 − 10a + 2.
After computation in Magma, we were able to prove that for each prime q ≡ 3 mod 4
the polynomial Jq has degree m(q), where m(s) is Kimberley’s formula.
24 / 34

Additional conjectures
In fact, our computations support these conjectures:
I For each prime q ≡ 3 mod 4, the polynomial Jq is precisely pq as computed by

PSLQ.
I For each integer s ≥ 1, ps is the unique degree m(s) prime factor of Js .
I The J function is a divisibility sequence: m | n implies Jm | Jn .
I For each positive integer s, both Js and ps have largest real root αs .

25 / 34

Proofs of Kimberley’s formula and the palindromic property
I In March 2016, DHB presented our results at a seminar at the University ofCalifornia, Berkeley.
I Following the presentation, Watson Ladd, a graduate student in mathematics,

brought to our attention the fact that some of our conjectures should follow fromresults in the theory of elliptic curves, Gaussian integers and ideals.
I After some effort, Ladd produced proofs of Kimberley’s formula and thepalindromic property, which proofs were then incuded in our paper and returnedto the journal.

26 / 34

Poisson polynomials: Progress and challenges
I Our computations employed up to 64,000-digit precision, producing polynomials
with degrees up to 512 and integer coefficients up to 10229 . These are the largest

successful integer relation computations to date.
I Kimberley’s formula was affirmed in every case x = y = 1/s, for s up to 52(except for s = 41, 43, 47, 49, 51), and also for s = 60 and s = 64.
I The resulting polynomial coefficients yielded clues that ultimately led to a proof ofKimberley’s formula and the palindromic property, employing techniques of elliptic
curves, Gaussian integers and ideals.
I Additional research is needed to understand many other combinations, e.g.,

x = p/s and y = q/s, for different values of p, q and s.
I A fundamentally new integer relation algorithm may be required to further extend

the requisite computations.
Full details are at:
http://www.davidhbailey.com/dhbpapers/poisson-res.pdf
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Jon Borwein’s research on π and experimental mathematics
Jon’s interest in π and experimental math wasprompted in part by his desire to do researchthat would connect to a large public audience.
π continues to excite millions worldwide,
leading many to pursue careers in math,science and engineering.
Experimental mathematics enables a muchbroader community to do real math research:
I High school and college students.
I Computer scientists.
I Computer graphics experts.
I Statisticians.
I Data scientists.
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Von Neumann’s warning about the future of mathematics
Experimental mathematics provides a means to escape the trap feared by John von
Neumann when he wrote,
But there is a grave danger that the subject [of mathematics] will develop
along the line of least resistance, that the stream, so far from its source, willseparate into a multitude of insignificant branches, and that the discipline will
become a disorganized mass of details and complexities. In other words, at agreat distance from its empirical source, or after much “abstract” inbreeding,a mathematical subject is in danger of degeneration. . . .
In any event, whenever this stage is reached, the only remedy seems to me tobe the rejuvenating return to the source: the re-injection of more or lessdirectly empirical ideas. I am convinced that this was a necessary condition
to conserve the freshness and the vitality of the subject and that this willremain equally true in the future.
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Winning the battle, but losing the war
Mathematicians and scientists may be winning battles to publish papers and obtain
grants, but we are losing the war for the hearts and minds of the public:
I 51% in U.S. (54% in Australia) either do not believe in climate change, or do not believe

there is any human connection.
I 42% in U.S. (23% in Australia) believe that humans were created within past 10,000 years.
I 38% in U.S. (32% in Australia) do not believe in evolution.
I 32% in U.S. (24% in Australia) do not believe vaccinations are safe.
I 48% in U.S. (34% in Australia) believe humans are being visited by extraterrestrial UFOs.
I 6% in U.S. believe NASA faked the Apollo moon landings.
I Some even dispute the value of π. (I frequently receive such email.)

Anti-science movements arise from both sides of the political spectrum:
I From the left: anti-vaccination and anti-fluoridation.
I From the right: anti-climate change and anti-evolution.
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Carl Sagan’s warning (The Demon Haunted World, 1995)
I have a foreboding of an America in my children’s or my grandchildren’s time — when
the United States is a service and information economy; when nearly all the key
manufacturing industries have slipped away to other countries; when awesome
technological powers are in the hands of a very few, and no one representing the public
interest can even grasp the issues; when the people have lost the ability to set theirown agendas or knowledgeably question those in authority; when, clutching ourcrystals and nervously consulting our horoscopes, our critical faculties in decline,
unable to distinguish between what feels good and what’s true, we slide, almost
without noticing, back into superstition and darkness. ...
We’ve arranged a global civilization in which most crucial elements ... profoundly
depend on science and technology. We have also arranged things so that almost no
one understands science and technology. This is a prescription for disaster. We might
get away with it for a while, but sooner or later this combustible mixture of ignorance
and power is going to blow up in our faces.
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How can we turn the tide?
I Start a blog.
I Visit schools or give lectures.
I Write books for the general public.
I Write articles for science news forums.
I Write expository articles for scientific journals.
I Pursue research topics that have potentially wide appeal.
I Recognize communication skills in hiring, promotion andresearch grant decisions.
I Find ways to utilize computers and otherwise make teaching

and research much more engaging and interesting.

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Ending the war between science and the humanities
Given the growing tensions in society, and the impact of rapidly changing
technology, we can no longer afford a war between the science-tech world
and the humanities:
I Those in math, science and technology must learn more about thehumanities, to better appreciate these fields, and to better
communicate to the public.
I Those in the humanities must learn more about math, science andtechnology, to better appreciate these fields, and to better participate
in dialogue on key issues.
It’s in Apple’s DNA that technology alone is not enough — thatit’s technology married with liberal arts, married with thehumanities, that yields us the result that makes our hearts sing.
[Steve Jobs]

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They should have sent a poet
In one memorable scene from the movie Contact, Jodi Foster
views a galaxy from her spacecraft, and is so overcome with
awe that she exclaims,
They should have sent a poet. So beautiful.
So beautiful... I had no idea.
In a similar way, those of us involved in research are often stunned by the beauty andelegance of mathematics and science, along with the rather mysterious fact that wehumans are able to comprehend these laws.
So why don’t we do more to share this wonder? Why don’t we write some poetry?
Thanks!
This talk is available here: http://www.davidhbailey.com/dhbtalks/dhb-jbcc-2017.pdf
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