This document outlines proofs for 8 previously unproven binary zero relations of degree 1. The relations include 5 length 24 relations involving base 212 and 3 length 40 relations involving base 220. The proofs were obtained through Fourier analysis of the normalized coefficient vectors, identifying real coefficients, and expressing the relations as linear combinations of simpler cosine and sine terms. This completes the proofs of all degree-1 binary zero relations currently listed in the compendium.

1. Degree 1 BBP-Type Zero Relations
Jaume Oliver Lafont
January 27, 2011
1 Introduction
Zero relations are BBP-type formulas that evaluate to zero. Although several
such formulas have been formally proved, others have remained unproved for
some time. This report outlines the proofs for eight previously unproved
binary zero relations.
2 Data
The compendium by D.H.Bailey includes eight zero relations found by PSLQ
but without any formal proof available. Five of them are length 24 base 212
relations, while the remaining three have length 40 and base 220 .
3 Methods
The series in the results section have been obtained through Fourier analysis
(written on PARI) of the normalized vector of coefficients from the P form
of the unproved zero relations. Vector coefficients are normalized in order to
cancel the 2n/2 factor. After identifying the real coefficients (using Plouffe’s
Inverter) simpler components have been obtained from the series, containing
only cosinus or sinus terms each.
WolframAlpha is able to check the correctness of the first five equations.
The remaining two are also easy to prove, for example, by splitting into sev-
eral terms and plugging each term into WolframAlpha. This step guarantees
that the identification of floating point coefficients is correct. After that, ob-
taining the integer coefficients from the series is easily done using two lines
of PARI. Equations 103, 105, 106, 107 and 108 in the compendium (version
January 21, 2011) can be written as linear combinations of equations 1, 2,
1

2. 3, 4 and 5 in this report. Similarly, equations 109, 110 and 111 in the com-
pendium can be written as linear combinations of equations 6 and 7 and 8
below. This completes the proofs.
4 Results
4.1 Length 24
∞ πn
1 − 2 cos 4 + cos (πn)
0=
n=1
n2n/2
=P (1, 24 , 8, (−8, 8, 4, 8, 2, 2, −1, 0))
=P (1, 212 , 24, (1)
(−2048, 2048, 1024, 2048, 512, 512, −256, 0,
− 128, 128, 64, 128, 32, 32, −16, 0,
− 8, 8, 4, 8, 2, 2, −1, 0))
∞ πn
1 − 2 cos 3 + cos (πn)
0=
n2n
n=1
=P (1, 26 , 6, (−16, 24, 8, 6, −1, 0))
∞ πn πn 5πn
1 − 2 cos 6 + 2 cos 2 − 2 cos 6 + cos (πn)
= (2)
n=1
n2n/2
=P (1, 212 , 24,
(0, −1024, 0, 1536, 0, 512, 0, 384, 0, −64, 0, 0,
0, −16, 0, 24, 0, 8, 0, 6, 0, −1, 0, 0))
∞ πn 7πn
sin 12 − sin 12
0=
n=1
n2n/2
=P (1, 212 , 24, (3)
(2048, −2048, −2048, 0, −512, −1024, −256, 0, −256, −128, 64, 0,
− 32, 32, 32, 0, 8, 16, 4, 0, 4, 2, −1, 0))
∞ πn
cos 12 − 2 cos πn + cos
4
7πn
12
0=
n=1
n2n/2
=P (1, 212 , 24, (4)
(2048, 0, −4096, −3072, −512, 0, 256, 768, 512, 0, −64, 0,
− 32, 0, 64, 48, 8, 0, −4, −12, −8, 0, 1, 0))
2

3. ∞ πn πn 7πn 5πn
1 sin 12 − 2 sin 6 + sin 12 + 2 sin 6
0 =√
3 n=1 n2n/2
=P (1, 212 , 24, (5)
(2048, −4096, 0, −1024, 512, 0, 256, 768, 0, 256, 64, 0,
− 32, −64, 0, −48, −8, 0, −4, 4, 0, 4, −1, 0))
4.2 Length 40
∞ πn
1 − 2 cos 4 + cos (πn)
0=
n=1
n2n/2
=P (1, 24 , 8, (−8, 8, 4, 8, 2, 2, −1, 0))
=P (1, 220 , 40,
(−524288, 524288, 262144, 524288, 131072, 131072, −65536, 0, (6)
− 32768, 32768, 16384, 32768, 8192, 8192, −4096, 0,
− 2048, 2048, 1024, 2048, 512, 512, −256, 0,
− 128, 128, 64, 128, 32, 32, −16, 0,
− 8, 8, 4, 8, 2, 2, −1, 0))
∞ πn πn 7πn 9πn 3πn 17πn
cos 20 − 3 cos 4 + cos 20 + cos 20 − cos 4 + cos 20
0=
n=1
n2n/2
=P (1, 220 , 40,
(524288, 0, −262144, −1310720, −786432, 0, 65536, 327680,
32768, 0, −16384, −8192, 0, −8192, 0, 24576, 20480,
2048, 0, −1024, 0, −512, 0, 256, 1280,
768, 0, −64, −320, −32, 0, 16, 80,
8, 0, −24, −20, −2, 0, 1, 0))
(7)
∞ πn πn 7πn 9πn 3πn 17πn
sin 20 − sin 4 − sin 20 + sin 20 + sin 4 + sin 20
0=
n=1
n2n/2
=P (1, 220 , 40,
(524288, −1572864, 262144, 0, 524288, 393216, −65536, 0,
(8)
32768, 65536, 16384, 0, −8192, 24576, 16384, 0,
2048, −6144, 1024, 0, −512, 1536, −256, 0,
− 512, −384, 64, 0, −32, −64, −16, 0,
8, −24, −16, 0, −2, 6, −1, 0))
3

4. 5 Conclusions
All degree-1 binary formulas currently listed in the Compendium are now proved.
4