Computer discovery of new mathematical facts and formulasDavid H. Bailey
http://www.davidhbailey.com
Lawrence Berkeley National Laboratory (retired)
University of California, Davis, Department of Computer Science

1 / 32

DHB receives a fax from the Simpsons
In October 1992, I received this fax from theSimpsons TV show, requesting the 40,000th
digit of π. I computed the first 40,000 digits,
and faxed the result back (noting that the
40,000th digit is a 1).
In the episode airing May 6, 1993, Apu, themanager of a convenience store, was challenged
by Marge’s attorney in a courtroom. He repliedthat he has a perfect memory. For example, hesaid, he can recite 40,000 digits of π, and thelast digit is a 1.

2 / 32

The PSLQ integer relation algorithm
Let (xn ) be a given vector of real numbers. An integer relation algorithm either finds
integers (an ) such thata1 x1 + a2 x2 + · · · + an xn = 0(to within the “epsilon” of the arithmetic being used), or else finds bounds within
which no relation can exist.
The “PSLQ” algorithm of mathematician-sculptor Helaman Ferguson (a Hamilton
College graduate) is the most widely used integer relation algorithm.
Integer relation detection requires very high precision (at least n × d digits, where d isthe size in digits of the largest ak ), both in the input data and in the operation of thealgorithm.
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), pg. 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), pg. 1719–1736.
3 / 32

Helaman Ferguson’s “Syzygy” sculpture at Hamilton College

4 / 32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 is the number of digits ofprecision).
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.
5 / 32

Decrease of log 10(min |yi |) in multipair PSLQ run

6 / 32

The binary algorithm for exponentiation
Problem: What is 317 mod 10?
Algorithm A:
317 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × ×3 = 129140163,
so answer = 3.
Algorithm B (faster): 317 = ((((32 )2 )2 )2 ) × 3 = 129140163, so answer = 3.
Algorithm C (fastest):
317 = ((((32 mod 10)2 mod 10)2 mod 10)2 mod 10) × 3 mod 10 = 3.
Note that in Algorithm C, we never have to deal with integers larger than 9 × 9 = 81.
In general, if reducing mod n, we never have to deal with integers larger than (n − 1)2 .

7 / 32

1996 result: How to compute arbitrary binary digits of log 2
Consider this well-known formula for log 2:
log 2 =

∞
X
1
n=1

n2n

= 0.101100010111001000010111111101111101000111001111011 . . .2

Note that the binary digits of log 2 beginning after position d can be written as{2d log 2}, where {·} denotes fractional part. Thus we can write:


∞
 X
d−n 
2
+
{2d log 2} =

n
n 
n=1
n=d+1


)
( d
∞
 X
X 2d−n mod n
2d−n 
( d)
X 2d−n

=

n=1

n

+



n=d+1

n 

,

where we have inserted modn since were are only interested in the fractional partwhen divided by n. Now note that the numerator 2d−n mod n can be calculated very
rapidly using the binary algorithm for exponentiation, using only standard arithmetic.
Is there a similar formula and scheme for π?

8 / 32

The BBP formula for π
In 1996, a PSLQ program discovered this new formula for π:
π =


∞
X
1
n=0

16k

4
2
1
1
−
−
−
.
8k + 1 8k + 4 8k + 5 8k + 6
This formula permits one to compute binary (or hexadecimal) digits of π beginning at
an arbitrary starting position.
This is the first known instance of a computer program discovering a new formula for π.
BBP-type formulas (also discovered using PSLQ) are now known for numerous othermathematical constants.

I D. H. Bailey, P. B. Borwein and S. Plouffe, “On the rapid computation of various polylogarithmic

constants,” Mathematics of Computation, vol. 66 (Apr 1997), 903–913.
9 / 32

Some other BBP-type
formulas found using PSLQ

∞
π2 =

1X 1
8
64k
k=0
∞

2 X 1
π =
27
729k
2

144
216
72
54
9
−
−
−
+
(6k + 1)2
(6k + 2)2
(6k + 3)2
(6k + 4)2
(6k + 5)2



k=0

−

243
405
81
27
−
−
−
(12k + 1)2(12k + 2)2(12k + 4)2(27k + 5)2

72
9
9
5
1
−
−
−
+
(12k + 6)2(12k + 7)2(12k + 8)2(12k + 10)2
(12k + 11)2∞

1 X 1
ζ(3) =
1792
212k
k=0







6144
43008
24576
30720
1536
−
+
+
−
(24k + 1)3(24k + 2)3(24k + 3)3(24k + 4)3(24k + 5)3

3072
768
3072
2688
192
1536
+
+
−
−
−
−
(24k + 6)3(24k + 7)3(24k + 9)3(24k + 10)3
(24k + 11)3
(24k + 12)3
96
672
384
24
48
12
−
−
−
+
+
−
(24k + 13)3
(24k + 14)3
(24k + 15)3
(24k + 17)3
(24k + 18)3
(24k + 19)3
+

120
48
42
3
+
−
+
(24k + 20)3
(24k + 21)3
(24k + 22)3
(24k + 23)3



I D. H. Bailey, J. M. Borwein, A. Mattingly and G. Wightwick, “The computation of previously

inaccessible digits of Pi2 and Catalan’s constant,” Notices of the AMS, vol. 60 (2013), 844–854. 10 / 32

Normal numbers
Given integer b ≥ 2, a real number x is b-normal (or “normal base b”) if every m-longstring of digits appears in the base-b expansion of x with limiting frequency 1/b m .
Using measure theory, it can be shown that almost all real numbers are b-normal for agiven integer base b (in fact, for all b simultaneously).
These are widely believed to be b-normal, for all integer bases b ≥ 2 and for all m ≥ 1:
I π = 3.14159265358979323846 . . .
I e = 2.7182818284590452354 . . .
√I
2 = 1.4142135623730950488 . . .
I log(2) = 0.69314718055994530942 . . .
I Every irrational algebraic number (this conjecture is due to Borel).
But there are no proofs of normality for any of these constants, not even for b = 2 andm = 1 (i.e., equal 0s and 1s in binary), much less for all b and all m.
Until recently, normality proofs were known only for a few contrived examples such as
Champernowne’s constant = 0.12345678910111213141516 . . . (which is 10-normal).
11 / 32

BBP formulas and normality
Consider a general BBP-type constant (a formula that permits the BBP technique):
α =

∞
X
p(n)
n=0

b n q(n),where p and q are integer polynomials, deg p < deg q, and q has no zeroes fornonnegative arguments.
In 2001, Richard Crandall and I proved that α is b-normal iff the sequence x0 = 0, and


p(n)
xn =
bxn−1 +,q(n)where {·} again denote fractional part, is equidistributed in the unit interval. Here“equidistributed” means that the sequence visits each subinterval (c, d) with limiting
frequency d − c.
I D. H. Bailey and R. E. Crandall, “On the Random Character of Fundamental Constant

Expansions,” Experimental Mathematics, vol. 10 (Jun 2001), 175–190.
12 / 32

Two specific examples
Consider the sequence x0 = 0 and
1xn = 2xn−1 +
.
n




Then log 2 is 2-normal iff this sequence is equidistributed in the unit interval.

Similarly, consider the sequence x0 = 0 and(xn =120n2 − 89n + 16
16xn−1 +512n4 − 1024n3 + 712n2 − 206n + 21).

Then π is 16-normal (and hence 2-normal) iff this sequence is equidistributed in theunit interval.

13 / 32

A class of provably normal constants
Crandall and I also proved that an infinite class of constants are 2-normal, e.g.
α2,3 =∞
Xn=1

1
3 n 2 3n

= 0.041883680831502985071252898624571682426096 . . .10= 0.0ab8e38f 684bda12f 684bf 35ba781948b0fcd6e9e0 . . .16
This constant was proven 2-normal by Stoneham in 1971, but we have extended this tothe case where (2, 3) are any pair (p, q) of relatively prime integers ≥ 2. We also
extended this result to an uncountable class: For any real r in [0, 1), the constant
α2,3 (r ) =

∞
X
n=1

1
3n 23n +rn

is 2-normal, where rn is the n-th bit in the binary expansion of r in [0, 1). These
constants are all distinct, so the class is uncountable.
I D. H. Bailey and R. E. Crandall, “Random Generators and Normal Numbers,” Experimental

Mathematics, vol. 11, no. 4 (2002), pg. 527–546.

14 / 32

“Random walk” on the first 100 billion base-4 digits of π

http://gigapan.com/gigapans/106803

15 / 32

“Random walk” on pseudorandom data

16 / 32

“Random walk” on α2,3 : Base 2 (normal) vs base 6 (nonnormal)

17 / 32

High-precision numerical integration using the tanh-sinh scheme
Given f (x ) defined on (−1, 1), define g(t) = tanh(π/2 sinh t). Seting x = g(t) yields
Z ∞

Z 1

f (x ) dx−1=−∞f (g(t))g 0 (t) dt ≈ hNX

wj f (xj ),j=−Nwhere xj = g(hj) and wj = g 0 (hj).
Features:
I Reducing h by half typically doubles the number of correct digits.
I Works well even for functions with blow-up singularities at endpoints.
I The cost of computing abscissas and weights increases only linearly with thenumber N of subdivisions, which is must faster than with most other methods.
1. D. H. Bailey, X. S. Li and K. Jeyabalan, “A comparison of three high-precision quadrature schemes,”
Experimental Mathematics, vol. 14 (2005), no. 3, pg. 317–329.
2. H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publications of RIMS,
Kyoto University, vol. 9 (1974), pg. 721-Ð741.
18 / 32

Ising integrals from mathematical physics
We applied our methods to study three classes of integrals: Cn are connected toquantum field theory, Dn arise in the Ising theory of mathematical physics, while the
En integrands are derived from Dn :
Cn :=

Dn :=

4
n!

Z ∞
0

0

4
n!

Z ∞

Z ∞

=

···

Z ∞

0

Z 1

En

···

2
0

0

···

Z 1
0



1
2
n
j=1 (uj + 1/uj )

P

Q

i<j



ui −ujui +uj

du1
dun
···
u1
un

2

P

n
j=1 (uj + 1/uj )

2

dun
du1
···
u1
un

2

uk − uj 
dt2 dt3 · · · dtnu+ uj
1≤j<k≤n kYwhere in the last line uk = t1 t2 · · · tk .
D. H. Bailey, J. M. Borwein and R. E. Crandall, “Integrals of the Ising class,” Journal of Physics A:
Mathematical and General, vol. 39 (2006), pg. 12271–12302.
19 / 32

Limiting value of Cn : What is this number?
Key observation: The Cn integrals can be converted to one- dimensional integrals
involving the modified Bessel function K0 (t):
Cn =

2n
n!

Z ∞
0

tK0n (t) dt

High-precision numerical values, computed using this formula and tanh-sinhquadrature, approach a limit. For example:
C1024 = 0.6304735033743867961220401927108789043545870787 . . .
What is this number? We copied the first 50 digits into the online Inverse Symbolic
Calculator (ISC) at https://isc.carma.newcastle.edu.au. The result was:lim Cn = 2e −2γ .

n→∞where γ denotes Euler’s constant. This is now proven.
20 / 32

Other Ising integral evaluations found using PSLQ
D3 = 8 + 4π 2 /3 − 27 L−3 (2)D4 = 4π 2 /9 − 1/6 − 7ζ(3)/2E2 = 6 − 8 log 2E3 = 10 − 2π 2 − 8 log 2 + 32 log2 2
E4 = 22 − 82ζ(3) − 24 log 2 + 176 log2 2 − 256(log3 2)/3+16π 2 log 2 − 22π 2 /3
E5 = 42 − 1984 Li4 (1/2) + 189π 4 /10 − 74ζ(3) − 1272ζ(3) log 2 + 40π 2 log2 2−62π 2 /3 + 40(π 2 log 2)/3 + 88 log4 2 + 464 log2 2 − 40 log 2
where ζ is the Riemann zeta function and Lin (x ) is the polylog function. E5 remained a“numerical conjecture” for several years, but was proven in March 2014 by Erik Panzer.

21 / 32

Using PSLQ to find the minimal polynomial of an algebraic number
One simple but important application of PSLQ is to recognize a computed numericalvalue 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 toa PSLQ program.
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
22 / 32

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 numbers x and y ,1φ2 (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.
23 / 32

Samples of minimal polynomials found by 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 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!
24 / 32

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?
25 / 32

Three improvements to the Poisson polynomial computation program

1. MPFUN-MPFR: 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
Total size of code: 190,000 lines

26 / 32

A fast algorithm to compute the Poisson potential function 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 = 0.0432139 . . ., and z = π2 (y + ix ). Compute the four theta functions
using the following 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
27 / 32

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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−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+2306104703709760221799424330422404147474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192-degree minimal polynomial found by multipair PSLQ for x = y = 1/35

This polynomial has degree 192, with coefficients as large as 1085 . This computation
required 18,000-digit arithmetic and 34 CPU-hours.
The case (1/37, 1/37) required 51,000-digit arithmetic and 90 CPU-days (5.6 days on
a 16-core parallel system).
Kimberley’s formula was upheld for (1/s, 1/s), for all s up to 52 (except fors = 41, 43, 47, 49, 51, which were too expensive), and also for s = 60 and s = 64.
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Palindromic polynomials
From our results, in the case (1/s, 1/s) where s is even, the resulting polynomial isalways 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.
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Proofs of Kimberley’s formula and the palindromic property
I On March 16, 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.
I The paper has now appeared:
David H. Bailey, Jonathan M. Borwein, Jason Kimberley and Watson Ladd,“Computer discovery and analysis of large Poisson polynomials,” Experimental
Mathematics, 27 Aug 2016.
I A preprint is available here:
http://www.davidhbailey.com/dhbpapers/poisson-res.pdf.

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The computer as an active agent of discovery
I Thirty years ago, computations were derided in the mathematical research world:“Real mathematicians don’t compute.”
I Nowadays, with Mathematica, Maple and Sage, almost any mathematical concept

can be explored computationally.
I A new generation of computer-literate mathematicians is now using these tools,

and many are writing their own programs.
I Hundreds of papers have been published with results discovered by computer.

Everyone from first-year students to senior faculty, from research mathematicians tocomputer scientists, can participate in world-class mathematical research.
Thanks! This talk is available at
http://www.davidhbailey.com/dhbtalks/dhb-hamilton-math.pdf.
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