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EULER'S AND THE TAXI CAB RELATIONS AND OTHER
NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF
TWO CUBED INTEGERS
VLADIMIR PLETSER
Abstract.
We show that Euler's relation and the Taxi-Cab relation are both
solutions of the same equation. General solutions of sums of two consecutive
cubes equaling the sum of two other cubes are calculated. There is an innite
number of relations to be found among the sums of two consecutive cubes and
the sum of two other cubes, in the form of two families. Their recursive and
parametric equations are calculated.
AMS 2010 Mathematics Subject Classication: Primary 11D25; Secondary 11B37
1.
Introduction
The remarkable relation
(1.1)
33+ 43+ 53= 63
among the cubes of four successive integers is often attributed to Euler, while in
fact it was already known to P. Bungus in the XVIth century [1, 2]. No other
similar relation can be found between cubes of four successive integers.
Another well-known relation involving two dierent sums of two cubes is
(1.2)
1729 = 93+ 103= 123+ 13
often call the taxi-cab number or taxi-cab relation and attributed to Indian math-
ematician Ramanujan after he mentioned in 1919 to fellow British mathematician
Hardy that this number is remarkable in the fact that it is the smallest integer that
can be written as the sum of two positive cubes in two ways (see historical account
in e.g., [3]). However, this relation was already mentioned by French mathemati-
cian Frenicle in the XVIIth century [2, 4]. Nevertheless, we will refer in this paper
to (1.1) and (1.2) as Euler's relation and Ramanujan's taxi-cab relation.
In fact, both relations can be deduced from a same equation, as we show in
this paper. It is simple to see that one can nd other taxi-cab numbers smaller
than Ramanujan's by transferring one term from left to right of (1.1), introducing
negative integers and yielding successively
91 = 33+ 43= 63+ (−5)3
(1.3)
152 = 33+ 53= 63+ (−4)3
(1.4)
189 = 43+ 53= 63+ (−3)3
(1.5)
and so on. Other taxi-cab numbers can be found by multiplying each relations (1.3)
to (1.5) by
k3
, i.e., the cube of any integer
k
. Sequences A001235 and A051347 in
Key words and phrases.
Sums of two consecutive cubes ; Equal sums of two cubes ; Taxi-Cab
number ; Euler's relation.
1
EULER'S AND THE TAXI CAB RELATIONS AND OTHER NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF TWO CUBED INTEGERS2
the OEIS [5] list all taxi-cab numbers for respectively only positive integers and for
positive and negative integers. Numerous mathematicians and authors have worked
on sums of cubes and equal sums of cubes. Excellent summaries and numerous
results can be found e.g., in [2, 6].
In this paper, our interest is in numbers that can be written as sums of two cubes
in at least two ways, one of them involving two consecutive cubes. In Section, 2
we show rst that Euler's and the Taxi-Cab relations are solutions of the same
equation. We calculate then the general case of the sum of two consecutive cubes
equal to the sum of two other cubes. In Section 3, we characterize two innite
families of solutions of the sum of two consecutive cubes equaling the sum of two
other cubes.
2.
General equation
We show rst that (1.1) and (1.2) are both solutions of a same equation. If one
observes that the rst term on the right hand side in (1.2) and (1.3) is three units
larger than the rst term on the left hand side, we can write
(2.1)
N=n3+ (n+ 1)3= (n+ 3)3+ (n+α)3
with
α
integer and
N
the positive integer that can be represented in (at least)
two ways by sums of two consecutive cubes and of two other cubes, one of which
possibly negative. Equation (2.1) yields the two general solutions
(2.2)
n=−3α2+ 8±p−3 (α4+ 8 (α3−6α2+ 13α+ 2))
6 (α+ 2)
which produces integer solutions for
α=−8
, giving
n+= 3
and
n−= 9
. Equation
(2.1) yields then respectively (1.3) and (1.2), showing that (2.1) yields both Euler's
relation and Ramanujan's taxi-cab number relation.
Let us consider now the general equation
(2.3)
N=n3+ (n+ 1)3= (n+a)3+ (n+b)3
with
a
and
b
integers,
a > 0
and
b < 0
. Solving for
n
, the third degree equation
dened by the second equality in (2.3) reduces to a second degree equation
(2.4)
3n2(a+b−1) + 3na2+b2−1+a3+b3−1= 0
whose discriminant reads
(2.5)
D= 3 (a−b)4−a4+b4+ (a−1)4+ (b−1)4+ 1
providing two real solutions for
D > 0
, namely
(2.6)
n=
−3a2+b2−1±r3(a−b)4−a4+b4+ (a−1)4+ (b−1)4+ 1
6 (a+b−1)
As
N
must be positive, we limit our search to
b < 0<a<|b|
and discrete solutions
of (2.3) or (2.4) are found, as shown in Table 1 for
n < 1000
, and arranged in
increasing order of
N
.
Note also that similar relations but with coecients having opposite signs are
obtained for negative values of
a
and for
a0=−b+ 1
,
b0=−a+ 1
, and
n0=−n−1
.
EULER'S AND THE TAXI CAB RELATIONS AND OTHER NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF TWO CUBED INTEGERS3
Table 1.
Values of
a
,
b
,
n+
,
n−
, solutions of (2.3) for
n < 1000
and
b < 0< a < |b|
a b n+n−N=n3+ (n+ 1)3= (n+a)3+ (n+b)3
3 -8 3
91 = 33+ 43= 63+ (−5)3
2 -7 4
189 = 43+ 53= 63+ (−3)3
3 -8 9
1729 = 93+ 103= 123+ 13
10 -39 18
12691 = 183+ 193= 283+ (−21)3
9 -38 32
68705 = 323+ 333= 413+ (−6)3
10 -39 36
97309 = 363+ 373= 463+ (−3)3
105 -194 46
201159 = 463+ 473= 1513+ (−148)3
32 -127 58
400491 = 583+ 593= 903+ (−69)3
64 -243 107
2484755 = 1073+ 1083= 1713+ (−136)3
73 -258 108
2554741 = 1083+ 1093= 1813+ (−150)3
32 -103 121
3587409 = 1213+ 1223= 1533+ 183
248 -481 121
3587409 = 1213+ 1223= 3693+ (−360)3
37 -192 123
3767491 = 1233+ 1243= 1603+ (−69)3
43 -168 163
8741691 = 1633+ 1643= 2063+ (−5)3
91 -360 163
8741691 = 1633+ 1643= 2543+ (−197)3
819 -1208 197
15407765 = 1973+ 1983= 10163+ (−1011)3
57 -128 235
26122131 = 2353+ 2363= 2923+ 1073
184 -597 235
26122131 = 2353+ 2363= 4193+ (−362)3
77 -208 301
54814509 = 3013+ 3023= 3783+ 933
120 -629 393
121861441 = 3933+ 3943= 5133+ (−236)3
120 -629 411
139361059 = 4113+ 4123= 5313+ (−218)3
393 -1178 438
168632191 = 4383+ 4393= 8313+ (−740)3
152 -793 481
223264809 = 4813+ 4823= 6333+ (−312)3
128 -511 490
236019771 = 4903+ 4913= 6183+ (−21)3
3225 -4274 528
295233841 = 5283+ 5293= 37533+ (−3746)3
148 -687 562
355957875 = 5623+ 5633= 7103+ (−125)3
2258 -3367 562
355957875 = 5623+ 5633= 28203+ (−2805)3
512 -1591 607
448404255 = 6073+ 6083= 11193+ (−984)3
777 -1952 633
508476241 = 6333+ 6343= 14103+ (−1319)3
190 -999 640
525518721 = 6403+ 6413= 8303+ (−359)3
442 -1767 804
1041378589 = 8043+ 8053= 12463+ (−963)3
It is seen also that three sums of two cubes are found for
n= 121,163,235,562.
Other relations are given in OEIS [5] Sequences A352133 to A352136 and cases
with three sums of two cubes are given in Sequences A352220 to A352225.
3.
Two Infinite Families
Figure 3.1 shows a plot of the couples
(n, n +a)
for
0< n ≤275
(data are from
OEIS [5] Sequences A352135, A352136, A352222, A352223, A352224, A352225).
Two families are clearly visible along two curves.
The rst top curve (or rst family) includes all couples
(n, n +a)
such that
η= (n+a)+(n+b) = 2n+a+b
are regularly increasing odd integers as shown
EULER'S AND THE TAXI CAB RELATIONS AND OTHER NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF TWO CUBED INTEGERS4
Figure 3.1.
Plot
n+a
vs
n
0.E+00
1.E+07
2.E+07
3.E+07
4.E+07
5.E+07
6.E+07
7.E+07
0.E+00 1.E+05 2.E+05 3.E+05 4.E+05 5.E+05 6.E+05 7.E+05 8.E+05
n+a
Table 2.
Values of
n
,
n+a
,
n+b
,
η= 2n+a+b
for rst and second families
First family Second family
i n n +a n +b η n n +a n +b η
1 3 6 -5 1 4 6 -3 3
2 46 151 -148 3 121 369 -360 9
3 197 1016 -1011 5 562 2820 -2805 15
4 528 3753 -3746 7 1543 10815 -10794 21
5 1111 10090 -10081 9 3280 29538 -29511 27
6 2018 22331 -22320 11 5989 65901 -65868 33
7 3321 43356 -43343 13 9886 128544 -128505 39
8 5092 76621 -76606 15 15187 227835 -227790 45
9 7403 126158 -126141 17 22108 375870 -375819 51
10 10326 196575 -196556 19 30865 586473 -586416 57
11 13933 293056 -293035 21 41674 875196 -875133 63
12 18296 421361 -421338 23 54751 1259319 -1259250 69
13 23487 587826 -587801 25 70312 1757850 -1757775 75
14 29578 799363 -799336 27 88573 2391525 -2391444 81
15 36641 1063460 -1063431 29 109750 3182808 -3182721 87
16 44748 1388181 -1388150 31 134059 4155891 -4155798 93
17 53971 1782166 -1782133 33 161716 5336694 -5336595 99
18 64382 2254631 -2254596 35 192937 6752865 -6752760 105
19 76053 2815368 -2815331 37 227938 8433780 -8433669 111
20 89056 3474745 -3474706 39 266935 10410543 -10410426 117
in Table 2 for the rst twenty cases, while for the second below curve (or second
family),
η= 2n+a+b
are regularly increasing odd multiples of
3
.
EULER'S AND THE TAXI CAB RELATIONS AND OTHER NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF TWO CUBED INTEGERS5
3.1.
Recursive relations.
The values of
n
,
n+a
and
n+b
of both the rst and
second families can be found by the recurrence relations
ni= 3ni−1−3ni−2+ni−3+κ
(3.1)
(n+a)i= 3 (n+a)i−1−3 (n+a)i−2+ (n+a)i−3+λ
(3.2)
(n+b)i= 3 (n+b)i−1−3 (n+b)i−2+ (n+b)i−3−λ
(3.3)
with
κ= 72
and
216
and
λ= 576 (i−2)
and
1728 (i−2)
for respectively the rst
and second families, and the rst three values of
ni
,
(n+a)i
and
(n+b)i
from
Table 2.
3.2.
Parametric relations.
We see from Table 2 that the fourth term
n+b
of
(2.3) is negative and is decreasing regularly with increasing
n
. So, let us pose
n+b=−(n+a) + β
, yielding from (2.3)
(3.4)
N=n3+ (n+ 1)3= (n+a)3−(n+a−β)3
For specic relations between
a
and
n
, one obtains two innite families of solutions
as shown in the following two theorems, giving parametric solutions for
n
,
N
,
n+a
and
n+b
.
Theorem 1.
For
∀i∈Z+
0
,
∃n, a, β ∈Z+
0
, such that
(3.5)
a= (β−1) n+β2+β+ 1
and an innite family of solutions of (3.4) exists for
β
odd,
(3.6)
β= 2i−1
yielding
n=
(2i−1) 3 (2i−1)2+ 4−1
2
(3.7)
N=
(2i−1) 3 (2i−1)2+ 4(2i−1)23 (2i−1)2+ 42
+ 3
4
(3.8)
n+a=
3 (2i−1)2(2i−1)2+ 2+ 2i+ 1
2
(3.9)
n+b=−
3 (2i−1)2(2i−1)2+ 2−2i+ 3
2
(3.10)
Proof.
Let
n, a, β, i ∈Z+
0
, and let
a
,
n
and
β
satisfy (3.5). Relation (3.4) yields
then the third degree equation
(3.11)
n3+ (n+ 1)3−βn +β2+β+ 13+βn +β2+ 13= 0
that simplies immediately in the product of a linear and a quadratic relations
(3.12)
2n−β3β2+ 4+ 1n2+ (2β+ 1) n+β2+β+ 1= 0
As the discriminant of the right quadratic polynomial is always negative, the qua-
dratic equation yields two complex solutions of no interest here. The right linear
equation yield the only real solution
(3.13)
n=β3β2+ 4+ 1
2
EULER'S AND THE TAXI CAB RELATIONS AND OTHER NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF TWO CUBED INTEGERS6
As
n
must be integer,
β
cannot be even and must be odd,
β= 2i−1
, yielding (3.7)
to (3.10).
Theorem 2.
For
∀i∈Z+
0
,
∃n, a, β ∈Z+
0
, such that
(3.14)
a=(β−3) n+ 2β
3
and an innite family of solutions of (3.4) exists for
β≡3
mod
6
,
(3.15)
β= 3 (2i−1)
yielding
n=9 (2i−1)3−1
2
(3.16)
N=
27 (2i−1)327 (2i−1)6+ 1
4
(3.17)
n+a=
3 (2i−1) 3 (2i−1)3+ 1
2
(3.18)
n+b=−
3 (2i−1) 3 (2i−1)3−1
2
(3.19)
Proof.
Let
n, a, β, i ∈Z+
0
, and let
a
,
n
and
β
satisfy (3.14). Relation (3.4) yields
then the third degree equation
(3.20)
n3+ (n+ 1)3−β
3(n+ 2)3
+β
3(n−1)3
= 0
that simplies immediately in the product of a linear and a quadratic relations
(3.21)
6n−β3+ 3n2+n+ 1
3= 0
The right quadratic equation yields two complex solutions of no interest here. The
right linear equation yield the only real solution
(3.22)
n=β3−3
6
As
n
must be integer,
β
must
3
mod
6
,
β= 3 (2i−1)
, yielding (3.16) to (3.19).
4.
Conclusion
We have shown rst that Euler's relation and the Taxi-Cab relation are both
solutions of the same equation. We have then calculated general solutions of sums
of two consecutive cubes equaling the sum of two other cubes. We have nally
shown that there is an innite number of relations that can be found among the
sums of two consecutive cubes and the sum of two other cubes, in the form of two
families and we have given their recursive and parametric equations.
Acknowledgment
The Author wishes to acknowledge the help of an OEIS Associate Editor and
Editor-in-Chief, for additional computing in OEIS [5] Sequence A352135.
EULER'S AND THE TAXI CAB RELATIONS AND OTHER NUMBERS THAT CAN BE WRITTEN TWICE AS SUMS OF TWO CUBED INTEGERS7
References
[1] Bungus P. (1591).
Numerorum Mysteria, 1618
, 463; Pars Altera, 65.
[2] Dickson L.E. (2005).
History of the Theory of Numbers, Vol. II: Diophantine Analysis
, Dover
Publications, New York, 550-562.
[3] Grinstein A. ( 2022).
Ramanujan and 1729
, University of Melbourne
Dept. of Math and Statistics Newsletter: Issue 3, 1998. available at
https://web.archive.org/web/20040320144821/http://zadok.org/mattandloraine/1729.html,
Last accessed 17 March 2022.
[4] Frenicle de Bessy B.(1657).
Commercium Epistolicum de Wallis, letter X
, Brouncker to Wallis,
Oct. 13, 1657.
[5] Sloane N.J.A., ed. (2022).
The On-Line Encyclopedia of Integer Sequences
, published elec-
tronically at https://oeis.org.
[6] Piezas III T. (2010).
A Collection of Algebraic Identities, Chap 6: Third Powers
, available at
https://sites.google.com/site/tpiezas/Home, Last accessed 2 April 2022.
European Space Agency (ret.)
Email address
:
Pletservladimir@gmail.com
