Search: a018786 -id:a018786
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A001235
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Taxi-cab numbers: sums of 2 cubes in more than 1 way.
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+10
114
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
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EXAMPLE
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4104 belongs to the sequence as 4104 = 2^3 + 16^3 = 9^3 + 15^3.
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MATHEMATICA
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Select[Range[750000], Length[PowersRepresentations[#, 2, 3]]>1&] (* Harvey P. Dale, Nov 25 2014, with correction by Zak Seidov, Jul 13 2015 *)
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PROG
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(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);
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CROSSREFS
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Solutions in greater numbers of ways:
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KEYWORD
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nonn,nice,changed
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AUTHOR
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STATUS
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approved
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A003824
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Numbers that are the sum of two 4th powers in more than one way (primitive solutions).
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+10
12
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635318657, 3262811042, 8657437697, 68899596497, 86409838577, 160961094577, 2094447251857, 4231525221377, 26033514998417, 37860330087137, 61206381799697, 76773963505537, 109737827061041, 155974778565937
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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D. J. Bernstein, sortedsums (contains software for computing this and related sequences)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A309762
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Numbers that are the sum of 3 nonzero 4th powers in more than one way.
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+10
10
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4, so 2673 is in the sequence.
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MAPLE
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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:
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MATHEMATICA
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Select[Range@350000, Length@Select[PowersRepresentations[#, 3, 4], ! MemberQ[#, 0] &] > 1 &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A309763
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Numbers that are the sum of 4 nonzero 4th powers in more than one way.
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+10
9
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4, so 259 is in the sequence.
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MAPLE
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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:
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MATHEMATICA
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Select[Range@20000, Length@Select[PowersRepresentations[#, 4, 4], ! MemberQ[#, 0] &] > 1 &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A016078
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Smallest number that is sum of 2 positive n-th powers in 2 different ways.
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+10
6
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OFFSET
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1,1
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COMMENTS
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If it exists, a(5) > 1.02*10^26 (see eqn. (27) at the Mathworld link). - Jon E. Schoenfield, Jan 05 2019
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LINKS
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FORMULA
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EXAMPLE
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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).
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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STATUS
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approved
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A230562
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Smallest number that is the sum of 2 positive 4th powers in >= n ways.
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+10
5
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OFFSET
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0,2
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COMMENTS
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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."
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REFERENCES
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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.
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LINKS
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EXAMPLE
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0 = (empty sum).
2 = 1^4 + 1^4.
635318657 = 59^4 + 158^4 = 133^4 + 134^4.
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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STATUS
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approved
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A255351
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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.
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+10
5
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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], ...
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PROG
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(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))))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A255352
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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.
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+10
4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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], ...
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PROG
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(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))]", "))))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A366703
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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.
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+10
1
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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), ...
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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