A202173 Conjectured list of numbers not of the form x^3 + y^3 + 2*z^3.

76, 148, 183, 356, 418, 428, 445, 491, 580, 671, 788, 931, 967, 1084, 1121, 1184, 1210, 1219, 1228, 1247, 1499, 1508, 1562, 1618, 1723, 1975, 2020, 2129, 2164, 2236, 2300, 2332, 2362, 2372, 2452, 2470, 2561, 2722, 2794, 2857

Offset

1,1

Comments

Wacław Sierpiński asked if a(1) = 76.

Searched -50000 <= x,y <= 50000 for terms through 3000.

Seiji Tomita, on the math-fun mailing list (May 18 2012), gives

1444 = -24062122787^3 - 9841546529^3 + 2*19524116332^3

1462 = 111091225^3 - 110862443^3 - 2*16168112^3

1588 = -6314285^3 - 6232583^3 + 2*6273700^3

2246 = -7194061^3 - 2344975^3 + 2*5775101^3

2822 = 8070731^3 - 3630235^3 - 2*6205213^3

Very little is known about this sequence; I do not know if any of these terms are correct. Can the sequence be proved nonempty? Theorem 3.3 in Broughan shows that modular arguments will not suffice. - Charles R Greathouse IV, Jun 12 2012

Hypothesis: the sequence is empty. See the Cohen book for details. - Arkadiusz Wesolowski, Aug 20 2013

References

Henri Cohen, Number Theory. Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics 239, Springer, 2007, p. 381.

Chao Ko: Decompositions into four cubes. Journ. London Math. Soc., 11 (1936), pp. 218-219.

L. J. Mordell: On the four integer cubes problem. Journ. London Math. Soc., 11 (1936), pp. 208-218.

Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, pp. 26, 72.

Links

Table of n, a(n) for n=1..40.

Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.

Eckford Cohen, Arithmetical functions associated with arbitrary sets of integers, Acta Arithmetica 5 (1959), pp. 407-415.

Andrzej Makowski, Sur quelques problèmes concernant les sommes de quatre cubes, Acta Arithmetica 5 (1959), pp. 121-123.

A. Schinzel and W. Sierpiński, Sur les sommes de quatre cubes, Acta Arithmetica 4 (1958), pp. 20-30.

Example

37 can be expressed as 2^3 + 3^3 + 2*1^3 or (-12)^3 + 13^3 + 2*(-6)^3, so 37 is not in the sequence.

Mathematica

Complement[Range[98], Flatten[Table[x^3 + y^3 + 2*z^3, {x, -35, 143}, {y, -62, 259}, {z, -209, 52}]]]

Crossrefs

Cf. A014136.

Sequence in context: A261705 A248532 A186896 * A004924 A004944 A063359

Adjacent sequences: A202170 A202171 A202172 * A202174 A202175 A202176

Keyword

nonn

Author

Charles R Greathouse IV and Arkadiusz Wesolowski, Dec 22 2011

Status

approved