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A332201
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Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| >= |y| >= |z|, or 0 if no such x exists.
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3
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0, 1, 1, 1, 0, 0, 2, 2, 2, 2, 2, 3, -11, 0, 0, 2, 2, 2, 3, 3, 3, 16, 0, 0, 2, 3, 3, 3, 3, 3, 2220422932, 0, 0
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OFFSET
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0,7
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COMMENTS
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It is known that there is no solution for n congruent to +-4 (mod 9), but it is now conjectured that there is a solution (and probably infinitely many such) for all other numbers. The numbers n = 0, 1 and 2 are the only cases for which infinite families of parametric solutions are known, for other n the solutions seem to be sporadic.
Search on this problem was motivated by a statement in Mordell's paper from 1953. Beck et al. found a solution for n = 30 in 1999, and for 52 in 2000. Huisman found a solution for n = 74 in 2016. A solution for 33 was found by Booker in 2019. The number 42 was the last one below 100 for which a solution was found, in late 2019, using a collaborative effort with supercomputers and home computers from volunteers.
For n < 30, we have a(n) = A246869(n+1) for the nonzero values, while A246869(n+1) = 2 for n == 4 or 5 (mod 9) up to there.
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LINKS
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B. Haran, Sum of Three Cubes - Numberphile, YouTube playlist (featuring videos from Nov 06 2015, May 31 2016 (about 74), Mar 12 2019 (about 33), Sep 2019 (about 42).
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FORMULA
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a(n) = 0 for n == 4 or n == 5 (mod 9).
a(n) <= k if |n - k^3| < 3 or |n - 2*k^3| < 2 or n = 3*k^3 for some k.
a(n) = A246869(n+1) for all n < 30 with a(n) > 0.
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EXAMPLE
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0 = 0^3 + 0^3 + 0^3, 1 = 1^3 + 0^3 + 0^3,
2 = 1^3 + 1^3 + 0^3, 3 = 1^3 + 1^3 + 1^3,
6 = 2^3 - 1^3 - 1^3, 7 = 2^3 - 1^3 + 0^3,
8 = 2^3 + 0^3 + 0^3, 9 = 2^3 + 1^3 + 0^3,
10 = 2^3 + 1^3 + 1^3, 11 = 3^3 - 2^3 - 2^3,
12 = -11^3 + 10^3 + 7^3, 15 = 2^3 + 2^3 - 1^3,
16 = 2^3 + 2^3 + 0^3, 17 = 2^3 + 2^3 + 1^3,
18 = 3^3 - 2^3 - 1^3, 19 = 3^3 - 2^3 + 0^3,
20 = 3^3 - 2^3 + 1^3, 21 = 16^3 - 14^3 - 11^3,
24 = 2^3 + 2^3 + 2^3, 25 = 3^3 - 1^3 - 1^3,
26 = 3^3 - 1^3 + 0^3, 27 = 3^3 + 0^3 + 0^3,
28 = 3^3 + 1^3 + 0^3, 29 = 3^3 + 1^3 + 1^3,
30 = 2220422932^3 - 2218888517^3 - 283059965^3 was discovered by Beck, Pine, Yarbrough and Tarrant in 1999 following an approach suggested by N. Elkies.
33 = 8866128975287528^3 - 8778405442862239^3 - 2736111468807040^3 was found by A. Booker in 2019. It is uncertain whether these are the smallest solutions.
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PROG
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(PARI) apply( A332201(n, L=oo)={!bittest(48, n%9)&& for(c=0, L, my(t1=c^3-n, t2=c^3+n, a); for(b=0, c, ((ispower(t1-b^3, 3, &a)&&abs(a)<=c)||(ispower(t1+b^3, 3, &a)&&abs(a)<=c))&&return(c); ispower(t2-b^3, 3, &a) && abs(a)<=c && return(-c)))}, [0..29])
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CROSSREFS
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KEYWORD
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sign,more,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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