A050790 -id:A050790 - cp's OEIS frontend

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

Patrick De Geest, Sep 15 1999

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

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

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.

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

Cf. A050788, A050789, A050790, A050791.

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

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 *)

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

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

More terms from Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Definition corrected by Robert Israel, Jun 09 2014

A050788 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of x.

Original entry on oeis.org

6, 71, 135, 372, 426, 242, 566, 791, 236, 575, 1938, 2676, 1124, 2196, 1943, 1851, 1943, 7676, 3318, 10866, 3086, 3453, 17328, 4607, 28182, 10230, 25765, 31212, 7251, 34199, 6560, 15218, 29196, 54101, 32882, 51293, 17384, 8999, 58462, 75263

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

From Fred W. Helenius (fredh(AT)ix.netcom.com), Jul 22 2008: (Start)
There is an infinite family of solutions to c^3+1=a^3+b^3 given by
(a,b,c) = (9n^3 + 1, 9n^4, 9n^4 + 3n). The present sequence actually asks about
x^3 + y^3 = z^3 - 1 with x < y < z; for that we can take
(x,y,z) = (9n^3 - 1, 9n^4 - 3n, 9n^4) for n > 1.
I extracted these solutions from Theorem 235 in Hardy & Wright; the result shown there is that all nontrivial rational solutions of
x^3 + y^3 = u^3 + v^3 are given by
x = r(1 - (a - 3b)(a^2 + 3b^2))
y = r((a + 3b)(a^2 + 3b^2) - 1)
u = r((a + 3b) - (a^2 + 3b^2)^2)
v = r((a^2 + 3b^2)^2 - (a - 3b))
where r,a,b are rational and r is not zero.
Specializing to r = 1, b = n/2 and a = 3n/2 gives
x = 1, y = 9n^3 - 1, u = 3n - 9n^4, v = 9n^4.
The solutions given above are obtained by changing signs and moving cubes from one side of the equation to the other as necessary.
Unfortunately, not all integral solutions are found so easily: the third value in A050788 corresponds to 135^3 + 138^3 = 172^3 - 1; this is not produced by such simple choices of r,a,b. (End)

			(575)^3 + 2292^3 = 2304^3 - 1.

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.

Jean-François Alcover, Table of n, a(n) for n = 1..60
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050787, A050789, A050790.

More terms from Jud McCranie, Dec 25 2000

Further terms from Don Reble, Nov 29 2001

A050789 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. The values of z (see A050787) are arranged in monotonically increasing order. Sequence gives values of y.

Original entry on oeis.org

8, 138, 138, 426, 486, 720, 823, 812, 1207, 2292, 2820, 3230, 5610, 5984, 6702, 8675, 11646, 11903, 16806, 17328, 21588, 24965, 27630, 36840, 31212, 37887, 33857, 34566, 49409, 46212, 59022, 66198, 66167, 56503, 69479, 64165, 78244, 89970

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			575^3 + 2292^3 = 2304^3 - 1.

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.

Jean-François Alcover, Table of n, a(n) for n = 1..60
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050787, A050788, A050790.

More terms from Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Edited by N. J. A. Sloane, Feb 22 2009

cp's OEIS Frontend

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.

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A050788 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of x.

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A050789 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. The values of z (see A050787) are arranged in monotonically increasing order. Sequence gives values of y.

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Crossrefs

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