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 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. 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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 EXAMPLE 2304 is in the sequence because 575^3 + 2292^3 = 2304^3 - 1. 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 *) PROG (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 CROSSREFS Cf. A050788, A050789, A050790, A050791. Sequence in context: A363478 A303147 A317354 * A017198 A281789 A134176 Adjacent sequences: A050784 A050785 A050786 * A050788 A050789 A050790 KEYWORD nonn,nice AUTHOR Patrick De Geest, Sep 15 1999 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 STATUS approved

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Last modified August 11 05:50 EDT 2024. Contains 375059 sequences. (Running on oeis4.)