A255351 -id:A255351 - cp's OEIS frontend

A018786 Numbers that are the sum of two 4th powers in more than one way.

635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752

Offset: 1

David W. Wilson

nonn

Since 4th powers are squares, this is a subsequence of A024508, the analog for squares. Sequence A001235 is the analog for third powers (taxicab numbers). Sequence A255351 lists max {a,b,c,d} where a^4 + b^4 = c^4 + d^4 = a(n), while A255352 lists the whole quadruples (a,b,c,d). - M. F. Hasler, Feb 21 2015

			a(1) = 59^4 + 158^4 = 133^4 + 134^4.
a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - _M. F. Hasler_, Feb 21 2015

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

Mia Muessig, Table of n, a(n) for n = 1..30000 (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
Mia Muessig, Julia code for finding general taxicab numbers
Eric Weisstein's World of Mathematics, Biquadratic Number.
Eric Weisstein's World of Mathematics, Diophantine Equation.

Subsequence of A003336 (and hence A004831) and A024508 (and hence A001481).

Cf. A001235, A003336, A003824, A309762.

Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* Jean-François Alcover, Jul 26 2011 *)

n=4;L=[];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(t",")))) \\ M. F. Hasler, Feb 21 2015

list(lim)=my(v=List()); for(a=134,sqrtnint(lim,4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a,4)+1,min(sqrtnint(lim-a4,4),a), my(t=a4+b^4); for(c=a+1,sqrtnint(lim,4), if(ispower(t-c^4,4), listput(v,t); break)))); Set(v) \\ Charles R Greathouse IV, Jul 12 2024

A weak lower bound: a(n) >> n^2. - Charles R Greathouse IV, Jul 12 2024

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.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Feb 21 2015

nonn

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

			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], ...

Mia Müßig, Table of n, a(n) for n = 1..120000
Mia Müßig, Julia code for finding general taxicab numbers

Cf. A018786, A255351, A366703.

{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))]","))))}

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.

Original entry on oeis.org

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

Offset: 1

Mia Muessig, Oct 17 2023

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.

			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), ...

Mia Müßig, Julia code for finding general taxicab numbers.

Cf. A018786, A255351, A255352.

A257298 Numbers whose cube is of the form a^5 + b^5 - c^5 with a >= b > 0 and c not in {a,b}.

Original entry on oeis.org

144, 3969, 4114, 4608, 17918, 18723, 34992, 44944, 53176, 75076, 127008, 131648, 147456, 163500, 171698, 206116, 235225, 347778, 450000, 462220, 573376, 599136, 611524, 660969, 715716, 927799, 943020, 964467, 986049, 999702

Offset: 1

M. F. Hasler, May 21 2015

nonn

Otherwise said, d-values of nontrivial solutions to a^5 + b^5 = c^5 + d^3.

			144^3 = 192^5 + 156^5 - 204^5,
3969^3 = 126^5 + 126^5 - 63^5,
4114^3 = 143^5 + 121^5 - 110^5,
4608^3 = 1536^5 + 1248^5 - 1632^5, ...

Cf. A000578, A000584, A020896, A256603, A255351.

Most terms computed by Giovanni Resta, May 25 2015

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A018786 Numbers that are the sum of two 4th powers in more than one way.

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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.

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PARI

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.

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A257298 Numbers whose cube is of the form a^5 + b^5 - c^5 with a >= b > 0 and c not in {a,b}.

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