A050787 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.

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

Offset: 1

Author

Patrick De Geest, Sep 15 1999

Keywords

• nonn
• nice

Comments

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

Examples

			2304 is in the sequence because 575^3 + 2292^3 = 2304^3 - 1.
		

References

• 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.

Links

• Jean-François Alcover and Charles R Greathouse IV, Table of n, a(n) for n = 1..104 (first 60 terms from Alcover)
• Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Crossrefs

Cf. A050788, A050789, A050790, A050791.

Programs

Maple

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

Mathematica

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 *)

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);
is(n)=n>8&&#select(v->min(v[1],v[2])>1,thue(T,n^3-1))>0 \\ Charles R Greathouse IV, Oct 26 2014

Extensions

More terms from Jud McCranie, Dec 25 2000

More terms from Don Reble, Nov 29 2001

Definition corrected by Robert Israel, Jun 09 2014