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A202173 Conjectured list of numbers not of the form x^3 + y^3 + 2*z^3. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified February 26 09:41 EST 2024. Contains 370338 sequences. (Running on oeis4.)