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A050787
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Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (0 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of z.
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8
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9, 144, 172, 505, 577, 729, 904, 1010, 1210, 2304, 3097, 3753, 5625, 6081, 6756, 8703, 11664, 12884, 16849, 18649, 21609, 24987, 29737, 36864, 37513, 38134, 38239, 41545, 49461, 51762, 59049, 66465, 68010, 69709, 71852, 73627, 78529
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OFFSET
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1,1
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COMMENTS
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n^3 - 1 is expressible as the sum of two distinct positive cubes. [corrected by Altug Alkan, Apr 11 2016]
The subsequence of primes in the sequence begins: 577, 38239, 69709. - Jonathan Vos Post, May 13 2010
Sequence is infinite. One subsequence is b (m) = 9 m^4 = {9, 144, 729, 2304, 5625, 11664, 21609, 36864, 59049, ...} = a (1, 2, 6, 10, 13, 17, 21, 24, 31, ...). - Zak Seidov, Sep 16 2013
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REFERENCES
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Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.
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LINKS
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EXAMPLE
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2304 is in the sequence because 575^3 + 2292^3 = 2304^3 - 1.
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MAPLE
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N:= 10000: # to get all entries <= N
P:= proc(r)
local dcands, xs;
dcands:= select(d -> issqr(-3*d^4+12*d*r), numtheory[divisors](r));
xs:= map(d -> [solve(d^2-3*d*x+3*x^2-r/d, x)], dcands);
select(p -> p[1]<>p[2], select(type, xs, list(posint)));
end proc:
select(z -> nops(P(z^3-1))>0, [$1..N]); # Robert Israel, Jun 09 2014
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MATHEMATICA
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r[z_] := Reduce[1 < x < y < z && x^3 + y^3 == z^3 - 1, {x, y}, Integers]; Reap[z = 4; While[z < 10^5, rz = r[z]; If[rz =!= False, Print[xyz = {x, y, z} /. ToRules[rz]]; Sow[xyz[[3]]]]; z++]][[2, 1]] (* Jean-François Alcover, Dec 27 2011, updated Feb 11 2014 *)
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PROG
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(PARI) is(n)=if(n<2, return(0)); my(c3=n^3); for(a=2, sqrtnint(c3-5, 3), if(ispower(c3-1-a^3, 3), return(1))); 0 \\ Charles R Greathouse IV, Oct 26 2014
(PARI) T=thueinit('x^3+1);
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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