A018786 -id:A018786

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A001235

Taxi-cab numbers: sums of 2 cubes in more than 1 way.

+10
114

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

From Wikipedia: "1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'"

A011541 gives another version of "taxicab numbers".

If n is in this sequence, then n*k^3 is also in this sequence for all k > 0. So this sequence is obviously infinite. - Altug Alkan, May 09 2016

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, Section D1.

G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.

Ya. I. Perelman, Algebra can be fun, pp. 142-143.

H. W. Richmond, On integers which satisfy the equation t^3 +- x^3 +- y^3 +- z^3, Trans. Camb. Phil. Soc., 22 (1920), 389-403, see p. 402.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165.

LINKS

Shahar Amitai, Table of n, a(n) for n = 1..30000 (terms a(1)-a(4724) from T. D. Noe, terms a(4725)-a(10000) from Zak Seidov).

Shahar Amitai, Python code to generate all taxicab numbers up to N.

J. Charles-É, Recreomath, Ramanujan's Number.

A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.

Henk Koppelaar, Peyman Nasehpour, and Maarten Looijen, Symmetry between Series if Entangled by Sums, Preprints.org, 2024.

Istanbul Bilgi University, Ramanujan and Hardy's Taxi

Christopher Lane, The First ten Ta(2) and their double distinct cubic sums representations, Find Ramanujan's Taxi Number using JavaScript. [WayBack Machine copy]

J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.

J. Loy, The Hardy-Ramanujan Number.

Mia Muessig, Julia code for finding general taxicab numbers

Ken Ono and Sarah Trebat-Leder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.

Eric Weisstein's World of Mathematics, Cubic Number

Eric Weisstein's World of Mathematics, Diophantine Equation 3rd Powers

Eric Weisstein's World of Mathematics, Taxicab Number

D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

EXAMPLE

4104 belongs to the sequence as 4104 = 2^3 + 16^3 = 9^3 + 15^3.

MATHEMATICA

Select[Range[750000], Length[PowersRepresentations[#, 2, 3]]>1&] (* Harvey P. Dale, Nov 25 2014, with correction by Zak Seidov, Jul 13 2015 *)

PROG

(PARI) is(n)=my(t); for(k=ceil((n/2)^(1/3)), (n-.4)^(1/3), if(ispower(n-k^3, 3), if(t, return(1), t=1))); 0 \\ Charles R Greathouse IV, Jul 15 2011

(PARI) T=thueinit(x^3+1, 1);

is(n)=my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 09 2016

CROSSREFS

Subsequence of A003325.

Cf. A007692, A008917, A011541, A018786, A018850 (primitive solutions), A051347 (allows negatives), A343708, A360619.

Solutions in greater numbers of ways:

(>2): A018787 (A003825 for primitive, A023050 for coprime),

(>3): A023051 (A003826 for primitive),

(>4): A051167 (A155057 for primitive).

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A003824

Numbers that are the sum of two 4th powers in more than one way (primitive solutions).

+10
12

635318657, 3262811042, 8657437697, 68899596497, 86409838577, 160961094577, 2094447251857, 4231525221377, 26033514998417, 37860330087137, 61206381799697, 76773963505537, 109737827061041, 155974778565937

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The prime divisors of elements of {a(n)} all appear to be in A045390. - David W. Wilson, May 28 2010

Conjecture: a(n) is congruent to 1,2,10 or 17 mod 24. - Mason Korb, Oct 07 2018

Wells selected a(1), with only about 12 other 9-digit numbers, for his Interesting Numbers book. - Peter Munn, May 14 2023

Dickson (1923) credited Euler with discovering 635318657 as a term, while Leech (1957) proved that it is the least term. - Amiram Eldar, May 14 2023

REFERENCES

L. E. Dickson, History of The Theory of Numbers, Vol. 2 pp. 644-7, Chelsea NY 1923.

R. K. Guy, Unsolved Problems in Number Theory, D1.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, p. 191.

LINKS

D. Wilson, Table of n, a(n) for n = 1..516 [The b-file was computed from Bernstein's list]

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4 = a(n)

D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)

D. J. Bernstein, sortedsums (contains software for computing this and related sequences)

Leonhard Euler, Resolutio formulae diophanteae ab(maa+nbb)=cd(mcc+ndd) per numeros rationales, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 13 (1802), pp. 45-63. See p. 47.

John Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.

Carlos Rivera, Puzzle 103. N = a^4+b^4 = c^4+d^4, The Prime Puzzles and Problems Connection.

E. Rosenstiel et al., The Four Least Solutions in Distinct Positive Integers of the Diophantine Equation s = x^3 + y^3 = z^3 + w^3 = u^3 + v^3 = m^3 + n^3, Instit. of Mathem. and Its Applic. Bull. Jul 27 (pp. 155-157) 1991.

Eric Weisstein's World of Mathematics, Diophantine equations, 4th powers

CROSSREFS

Cf. A018786.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson, Aug 15 1996

STATUS

approved

A309762

Numbers that are the sum of 3 nonzero 4th powers in more than one way.

+10
10

2673, 6578, 16562, 28593, 35378, 42768, 43218, 54977, 94178, 105248, 106353, 122018, 134162, 137633, 149058, 171138, 177042, 178737, 181202, 195122, 195858, 198497, 216513, 234273, 235298, 235553, 264113, 264992, 300833, 318402, 318882, 324818, 334802, 346673

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Robert Israel, Table of n, a(n) for n = 1..1616

Eric Weisstein's World of Mathematics, Biquadratic Number

EXAMPLE

2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4, so 2673 is in the sequence.

MAPLE

N:= 10^6: # for terms <= N

V:= Vector(N, datatype=integer[4]):

for a from 1 while a^4 <= N do

for b from 1 to a while a^4+b^4 <= N do

for c from 1 to b do

v:= a^4+b^4+c^4;

if v > N then break fi;

V[v]:= V[v]+1

od od od:

select(i -> V[i]>1, [$1..N]); # Robert Israel, Aug 19 2019

MATHEMATICA

Select[Range@350000, Length@Select[PowersRepresentations[#, 3, 4], ! MemberQ[#, 0] &] > 1 &]

CROSSREFS

Cf. A003337, A008917, A018786, A024796, A193243, A309763, A344192, A344239, A345010.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 15 2019

STATUS

approved

A309763

Numbers that are the sum of 4 nonzero 4th powers in more than one way.

+10
9

259, 2674, 2689, 2754, 2929, 3298, 3969, 4144, 4209, 5074, 6579, 6594, 6659, 6769, 6834, 7203, 7874, 8194, 8979, 9154, 9234, 10113, 10674, 11298, 12673, 12913, 13139, 14674, 14689, 14754, 16563, 16578, 16643, 16818, 17187, 17234, 17299, 17314, 17858, 18963, 19699

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Biquadratic Number

EXAMPLE

259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4, so 259 is in the sequence.

MAPLE

N:= 10^5: # for terms <= N

V:= Vector(N, datatype=integer[4]):

for a from 1 while a^4 <= N do

for b from 1 to a while a^4+b^4 <= N do

for c from 1 to b while a^4 + b^4+ c^4 <= N do

for d from 1 to c do

v:= a^4+b^4+c^4+d^4;

if v > N then break fi;

V[v]:= V[v]+1

od od od od:

select(i -> V[i]>1, [$1..N]); # Robert Israel, Oct 07 2019

MATHEMATICA

Select[Range@20000, Length@Select[PowersRepresentations[#, 4, 4], ! MemberQ[#, 0] &] > 1 &]

CROSSREFS

Cf. A000583, A003338, A018786, A025367, A025406, A309762, A344193, A344238, A344241, A344644.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 15 2019

STATUS

approved

A016078

Smallest number that is sum of 2 positive n-th powers in 2 different ways.

+10
6

4, 50, 1729, 635318657

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If it exists, a(5) > 1.02*10^26 (see eqn. (27) at the Mathworld link). - Jon E. Schoenfield, Jan 05 2019

LINKS

Table of n, a(n) for n=1..4.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers

FORMULA

If A230561(n) exists, then a(n) <= A230561(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3. - Jonathan Sondow, Oct 24 2013 [Comment edited by N. J. A. Sloane, Apr 03 2021]

EXAMPLE

4 = 1^1 + 3^1 = 2^1 + 2^1;

50 = 1^2 + 7^2 = 5^2 + 5^2,

1729 = 1^3 + 12^3 = 9^3 + 10^3;

635318657 = 59^4 + 158^4 = 133^4 + 134^4 = A018786(1).

MATHEMATICA

(* This is just an empirical verification *) Do[max = 4 + n^4; Clear[cnt]; cnt[_] = 0; smallest = Infinity; Do[ cnt[an = x^n + y^n] += 1; If[cnt[an] == 2 && an < smallest, smallest = an], {x, 1, max}, {y, x, max}]; Print["a(", n, ") = ", smallest], {n, 1, 4}] (* Jean-François Alcover, Aug 13 2013 *)

CROSSREFS

Cf. A046881, A230561.

KEYWORD

nonn,nice,hard,more

AUTHOR

Robert G. Wilson v, Dec 11 1999

STATUS

approved

A230562

Smallest number that is the sum of 2 positive 4th powers in >= n ways.

+10
5

0, 2, 635318657

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Hardy and Wright say that a(3) is unknown.

Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums."

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, 2008; section 21.11.

LINKS

Table of n, a(n) for n=0..2.

Carlos Rivera, Puzzle 103: N = a^4 + b^4 = c^4 + d^4

EXAMPLE

0 = (empty sum).

2 = 1^4 + 1^4.

635318657 = 59^4 + 158^4 = 133^4 + 134^4.

CROSSREFS

Cf. A003824, A016078, A018786, A046881, A088687, A088848, A088849, A088867, A230561.

KEYWORD

hard,more,nonn

AUTHOR

Jonathan Sondow, Oct 25 2013

STATUS

approved

A255351

Values of b = max {a,b,c,d} for solutions to a^4 + b^4 = c^4 + d^4, a < c < d < b, ordered by size of b.

+10
5

158, 239, 292, 316, 474, 478, 502, 542, 584, 631, 632, 717, 790, 876, 948, 956, 1004, 1084, 1106, 1168, 1195, 1203, 1262, 1264, 1381, 1422, 1434, 1460, 1506, 1580, 1626, 1673, 1738, 1752, 1893, 1896, 1912

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A018786 for the values of a^4 + b^4 = c^4 + d^4, and A255352 for the list of the full quadruples (a,b,c,d). See there for further comments, motivation and references.

The values of b listed here allow one to reproduce the full solutions (a,b,c,d) with not too much effort, cf. the inner loops of the PARI code.

LINKS

Mia Muessig, Table of n, a(n) for n = 1..30000

Mia Muessig, Julia code for finding general taxicab numbers

EXAMPLE

The quadruples [a,b,c,d] are, listed in order of increasing b = max{a,b,c,d}):

[59, 158, 133, 134], [7, 239, 157, 227], [193, 292, 256, 257], [118, 316, 266, 268], [177, 474, 399, 402], [14, 478, 314, 454], [271, 502, 298, 497], [103, 542, 359, 514], [386, 584, 512, 514], [222, 631, 503, 558], [236, 632, 532, 536], [21, 717, 471, 681], [295, 790, 665, 670], [579, 876, 768, 771], [354, 948, 798, 804], [28, 956, 628, 908], ...

PROG

(PARI) {n=4; for(b=1, 1999, for(a=1, b, t=a^n+b^n; for(c=a+1, sqrtn(t\2, n), ispower(t-c^n, n)||next; print1(b", "); next(3))))}

CROSSREFS

Cf. A018786, A255352.

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 21 2015

STATUS

approved

A255352

List of quadruples (a,b,c,d) with a^4 + b^4 = c^4 + d^4, a < c < d < b, listed in order of the largest term b.

+10
4

59, 158, 133, 134, 7, 239, 157, 227, 193, 292, 256, 257, 118, 316, 266, 268, 177, 474, 399, 402, 14, 478, 314, 454, 271, 502, 298, 497, 103, 542, 359, 514, 386, 584, 512, 514, 222, 631, 503, 558, 236, 632, 532, 536, 21, 717, 471, 681, 295, 790, 665, 670, 579, 876, 768, 771

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The Ramanujan taxicab number 1729 = 1^3 + 12^3 = 9^3 + 10^3 satisfies the equation a^n + b^n = c^n + d^n for n=3. The present sequence corresponds to the same equation with exponent n=4.

As far as is known, the existence of solutions to the equation with exponent n=5 remains an open question.

See A018786 for the values of a^4 + b^4 = c^4 + d^4. See A255351 for the list of b-values, which are sufficient to reconstruct the quadruples (cf. inner loops of the PARI code).

See A366703 for the quadruples which consist only of prime numbers. - Mia Muessig, Oct 23 2023

LINKS

Mia Müßig, Table of n, a(n) for n = 1..120000

Mia Müßig, Julia code for finding general taxicab numbers

EXAMPLE

The quadruples [a,b,c,d] are, listed in order of increasing b = max{a,b,c,d}):

PROG

(PARI) {n=4; for(b=1, 999, for(a=1, b, t=a^n+b^n; for(c=a+1, sqrtn(t\2, n), ispower(t-c^n, n)||next; print1([a, b, c, round(sqrtn(t-c^n, n))]", "))))}

CROSSREFS

Cf. A018786, A255351, A366703.

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 21 2015

STATUS

approved

A366703

List of quadruples (a,b,c,d) with a^4 + b^4 = c^4 + d^4, a < c < d < b, a,b,c,d prime, listed in order of the largest term b.

+10
1

7, 239, 157, 227, 40351, 62047, 46747, 59693

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A255352 for quadruples which do not necessarily consist of prime numbers. There are infinitely many such quadruples, because if (a, b, c, d) is in the sequence, so is (m*a, m*b, m*c, m*d). It is unknown whether there are infinitely many quadruples which consist only of prime numbers. The two given quadruples are the only ones with a^4 + b^4 = c^4 + d^4 <= 10^24.

LINKS

Table of n, a(n) for n=1..8.

Mia Müßig, Julia code for finding general taxicab numbers.

EXAMPLE

The quadruples (a,b,c,d), listed in order of increasing b = max{a,b,c,d}, are

(7, 239, 157, 227),

(40351, 62047, 46747, 59693), ...

CROSSREFS

Cf. A018786, A255351, A255352.

KEYWORD

nonn,hard,more

AUTHOR

Mia Muessig, Oct 17 2023

STATUS

approved

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